Semiconductor material phonon average free time spectrum reconstruction method and device and storage medium

Abstract

The invention relates to a semiconductor material phonon average free time spectrum reconstruction method and device and a storage medium. The method comprises the steps that S1, response temperature rise changes of a sample under excitation of alternating current heat flows with different heating frequencies omega H are obtained through experimental testing; s2, according to the first-order spherical harmonic function, approximately solving a Boltzmann transport equation to obtain a diffusion type equation, and in combination with a correction slip boundary condition, obtaining an analytical expression of theoretical temperature change of the sample, the theoretical temperature change of the sample is related to the amplitude and frequency of alternating current heat flow excitation, the size of a heater, the average free time spectrum of intrinsic phonons of the sample and the phonon penetration coefficient between the heater and the sample; and S3, after the experimental conditions are substituted, fitting the response temperature rise change and the theoretical temperature change measured by the experiment to obtain the average free time spectrum of the intrinsic phonons of the sample and the phonon penetration coefficient between the heater and the sample to complete the reconstruction of the average free time spectrum of the phonons. Compared with the prior art, the method has the advantages of high speed, high accuracy and the like.

Description

Technical Field

[0001] The present invention relates to the technical field of microscale thermophysical property detection, and in particular to a method, device and storage medium for reconstructing the mean free time spectrum of phonons of semiconductor materials based on the combination of ballistic transport theory and micro-nanoscale high-frequency testing experiments. Background Art

[0002] At present, the characteristic line width of the chip has been reduced to the nanometer level (nm), and the main frequency has exceeded the gigahertz (GHz). The thermal reliability of high-power density devices under ultra-high frequency conditions faces huge challenges. Thermal conduction in semiconductor materials mainly depends on phonons (i.e., the quantization of lattice vibrations). The study of its heat transport mechanism is particularly critical for understanding the thermal conductivity of nanostructured materials or materials in ultrafast heating processes, and has important guiding significance for chip thermal design and thermal management.

[0003] The intrinsic thermal conductivity of a material is described by the kinetic theory of gases as: Among them, C ω is the specific heat per unit volume of a phonon with a vibration angular frequency of ω, v ω is the phonon group velocity, Λ ω is the mean free path (MFP) of phonons, i.e., the statistical average of the distances traveled by two collisions of phonons. To more intuitively show the thermal conductivity of phonons with different mean free paths, the above formula can be transformed into: in The mean free path is Λ ω The contribution of phonons to thermal conductivity. It can be seen that the thermal conductivity of phonons presents a mean free path spectrum distribution, and the intrinsic thermal conductivity of the material is the integral sum of the thermal conductivity of phonons with different mean free paths. Reconstructing the spectral distribution of phonon thermal conductivity based on micro- and nano-scale test experiments is the basis for studying the mechanism of phonon heat transport and is also a research hotspot in the field of microscale heat transfer.

[0004] At present, the transition of phonon transport mode from diffusion transport to ballistic transport is achieved by regulating the characteristic size of the system (sample size, heater size). Under diffusion transport, phonons undergo sufficient scattering (collision), the mean free path is the eigenvalue, and the thermal conductivity is not affected. However, under ballistic transport, the phonon mean free path is suppressed by the sample size and will be reduced, and the thermal conductivity is limited. The same characteristic size has different degrees of suppression on the thermal conductivity of phonons with different mean free paths. By experimentally measuring the difference between the equivalent thermal conductivity and the bulk eigenvalue of the sample under different characteristic sizes, the phonon mean free path spectrum can be reconstructed. The theoretical research tools describing phonon transport mainly include molecular dynamics simulation (MD), first-principles calculation, Boltzmann transport equation (BTE), ballistic-diffusion equation (BDE), etc.; experimental testing methods mainly include 3ω test method, time domain (or frequency domain) laser thermoreflectance method (TDTR / FDTR), transient thermal grating (TTG), laser heating + X-ray detection, etc.

[0005] There are two problems with current technology:

[0006] 1. The above theoretical numerical calculation method is complex and requires a lot of computing resources and time. It uses the basic information of phonons as input to forward calculate the thermal conductivity, temperature and heat flow distribution of the sample, but cannot obtain a clear analytical solution of the physical image, and therefore cannot directly extract the phonon spectrum from the experimental data.

[0007] 2. The thermal models of the above experimental methods for testing acoustic trajectory transport are all based on the diffusion equation. The experimental data containing the acoustic trajectory transport phenomenon are fitted to the physical model describing diffusion transport, which causes the inconsistency between the experimental method and the tested physical phenomenon, thus affecting the accuracy of the reconstruction results. Summary of the invention

[0008] The purpose of the present invention is to provide a method, device and storage medium for reconstructing the mean free time spectrum of phonons in semiconductor materials.

[0009] The purpose of the present invention can be achieved by the following technical solutions:

[0010] A method for reconstructing a semiconductor material phonon mean free time spectrum, comprising:

[0011] Step S1: Get different heating frequencies ω through experimental testing H The response temperature rise of the sample under the AC heat flow excitation;

[0012] Step S2: Solving the Boltzmann transport equation according to the first-order spherical harmonic function approximation to obtain a diffusion equation, and combining it with the modified slip boundary condition to obtain an analytical expression for the theoretical temperature change of the sample, wherein in the analytical expression, the theoretical temperature change of the sample is related to the amplitude and frequency of the AC heat flow excitation, the size of the heater, the mean free time spectrum of the intrinsic phonons of the sample, and the phonon penetration coefficient between the heater and the sample;

[0013] Step S3: After substituting the measured temperature change of the sample, the amplitude and frequency of the AC heat flux excitation, and the size of the heater, the experimentally measured response temperature rise change and the theoretical temperature change are fitted to obtain the sample's intrinsic phonon mean free time spectrum and the phonon penetration coefficient between the heater and the sample to complete the reconstruction of the phonon mean free time spectrum.

[0014] The analytical expression for the theoretical temperature change of the sample is:

[0015]

[0016] Where: P0 is the amplitude of AC heat flow excitation, l is the length of the micro heater, C ω is the specific heat per unit volume of phonons with a phonon frequency of ω, v ω is the phonon group velocity, ∈ is the phonon penetration coefficient between the heater and the sample, λ is the Fourier cosine inverse transform integral variable, b is the half-width of the microheater, is the first intermediate parameter, H is the Heaviside step function, and t is time.

[0017] The amplitude of the AC heat flux excitation is:

[0018]

[0019] Where: I 1ω is the amplitude of the heating current, R e is the resistance of the heater.

[0020] The first intermediate parameter is:

[0021]

[0022] Where: k AC,ω is the AC thermal conductivity, is the second intermediate parameter.

[0023] The second intermediate parameter is:

[0024]

[0025] Where: H is the heating frequency, α AC,ω is the AC thermal diffusion coefficient.

[0026] The AC thermal conductivity is:

[0027]

[0028] Where: τ ω is the average free time.

[0029] The modified slip boundary condition is:

[0030]

[0031] Where: q net,bdy (x, y = 0) is the input excitation heat flux distribution at the upper surface boundary of the sample, G ω (y=∞) is the incident radiation energy of phonons at the bottom boundary of the sample, ε ω is the penetration coefficient of phonons with a phonon frequency of ω between the heater and the sample, is the unit normal vector at the boundary.

[0032] The step S3 comprises:

[0033] Step S3-1: obtaining the real part and the imaginary part of the theoretical temperature change expression, and obtaining the theoretical phase lag based on the real part and the imaginary part of the theoretical temperature change expression;

[0034] Step S3-2: Substituting the measured temperature change of the sample, the amplitude and frequency of the AC heat flow excitation, and the size of the heater, the real part and the imaginary part of the measured temperature change expression are obtained, and the measured phase lag is obtained based on the real part and the imaginary part of the measured temperature change expression;

[0035] Step S3-3: fitting the measured phase lag to the theoretical phase lag to obtain the phonon scattering coefficient and the contribution of phonons of each mean free path to the thermal conductivity;

[0036] Step S3-4: Obtain a reconstructed phonon mean free time spectrum based on the phonon scattering coefficient and the contribution of phonons of each mean free path to the thermal conductivity.

[0037] The contribution of the mean free path of phonons to the thermal conductivity is:

[0038]

[0039] Where: K τ is the phonon MFT spectrum function, indicating that the mean free time is τ ω The thermal conductivity of phonons.

[0040] A device for reconstructing the mean free time spectrum of phonons in semiconductor materials comprises a memory, a processor, and a program stored in the memory, wherein the processor implements the above method when executing the program.

[0041] A storage medium stores a program thereon, wherein the program implements the above method when executed.

[0042] Compared with the prior art, the present invention has the following beneficial effects: solving the defects of the existing theoretical numerical calculation methods, which have high requirements on computing resources, long calculation time, and cannot directly extract phonon information from experimental data, as well as the problem that the experimental method in the traditional phonon spectrum reconstruction technology is inconsistent with the tested physical phenomenon. The theoretical model of the present invention accurately and efficiently simulates the temperature and heat flow distribution in the sample with a concise analytical expression, which can be combined with various existing AC heating experimental methods to accurately, efficiently and self-consistently combine the experimental data and reconstruct the sample intrinsic phonon mean free time (MFT) spectrum and the phonon penetration coefficient between the heater and the sample. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 Schematic diagram of the technical route of the present invention, wherein (a) is a theoretical model, (b) is an experimental test, and (c) is a phonon MFT spectrum;

[0044] Figure 2 Schematic diagram of phonon mean free time (MFT) spectrum and its cumulative function;

[0045] Figure 3 A flowchart of the framework on which the present invention relies;

[0046] Figure 4 It is a schematic diagram of reconstructing the phonon MFT spectrum and phonon penetration coefficient by combining the present invention with the traditional 3omega test method;

[0047] Figure 5 The figure is a schematic flow chart of the main steps of the method of the present invention. DETAILED DESCRIPTION

[0048] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0049] A method for reconstructing the mean free time spectrum of phonons in semiconductor materials, such as Figure 5 As shown, including:

[0050] Step S1: Get different heating frequencies ω through experimental testing H The response temperature rise of the sample under the AC heat flow excitation;

[0051] Step S2: Solving the Boltzmann transport equation according to the first-order spherical harmonic function approximation to obtain a diffusion equation, and combining it with the modified slip boundary condition to obtain an analytical expression for the theoretical temperature change of the sample, wherein in the analytical expression, the theoretical temperature change of the sample is related to the amplitude and frequency of the AC heat flow excitation, the size of the heater, the mean free time spectrum of the intrinsic phonons of the sample, and the phonon penetration coefficient between the heater and the sample;

[0052] Step S3: After substituting the measured temperature change of the sample, the amplitude and frequency of the AC heat flux excitation, and the size of the heater, the experimentally measured response temperature rise change and the theoretical temperature change are fitted to obtain the sample's intrinsic phonon mean free time spectrum and the phonon penetration coefficient between the heater and the sample to complete the reconstruction of the phonon mean free time spectrum.

[0053] Specifically, the present application relates to a theoretical model of an AC heating experimental test method. In order to ensure the accuracy, efficiency and self-consistency of the theoretical model with experimental observation phenomena, the present application abandons the thermal diffusion model (diffusion equation) used to describe the traditional experimental test method and adopts the more basic Boltzmann transport equation (BTE) to obtain a temperature analytical expression ΔT that can describe both diffusion transport and ballistic transport. 理论 This expression is consistent with the heating frequency ω H , the sample phonon spectrum function (i.e., the contribution of each phonon to the thermal conductivity of the material), and the phonon penetration coefficient between the heater and the sample. The experimentally measured heater temperature ΔT 实验 Fit to the model to solve for temperature ΔT 理论 , the phonon spectrum can be extracted.

[0054] The present application involves AC heating experimental test methods including but not limited to the existing commonly used 3ω test method and time domain (or frequency domain) laser thermal reflection method (TDTR / FDTR). In order to reduce the complexity of processing samples of different sizes or adjusting the size of the heater, the present invention abandons the traditional phonon spectrum reconstruction technology to suppress the intrinsic thermal conductivity of the sample by regulating the characteristic size (sample size, heater size), and uses the material equivalent thermal conductivity and temperature rise changes under different characteristic times (AC heating frequencies) to extract the phonon mean free time (MFT) spectrum.

[0055] Theoretical model part, such as Figure 1(a), based on the Boltzmann transport equation (BTE) and the first-order spherical harmonic function approximation (P1 approximation), the present invention proposes the concept of AC thermal conductivity, transforms the complex integral-differential BTE equation into a heat diffusion type equation that can describe both diffusion transport and ballistic transport, and can directly map the solution of the existing classical diffusion equation into the BTE general solution form, and obtain the heater temperature ΔT by modifying the boundary conditions. 理论 The analytical expression of .

[0056] Experimental test part, such as Figure 1 As shown in (b), in the first step, a frequency of ω is applied to the sample to be tested through a heater. H The first step is to conduct AC heat flux excitation (heating methods include but are not limited to resistance heating in the 3ω test method and pump laser heating in the time domain (or frequency domain) laser thermal reflection method (TDTR / FDTR)); the second step is that the phonons of different MFTs in the sample to be tested respond differently to the AC thermal excitation, which is reflected in the temperature rise and heat flow changes in the sample, where the temperature rise is closely related to the phonon spectrum of the sample to be tested; the third step is to collect the sample temperature rise response signal. The surface temperature rise of the sample is easy to obtain, so it is the first choice for the temperature rise signal (detection methods include but are not limited to the average temperature rise of the resistor in the 3ω test method and the detection laser in the time domain (or frequency domain) laser thermal reflection method (TDTR / FDTR)); the fourth step is to repeat the operations from the first step to the third step, and change the heating frequency G from small to large. H The phonons of different MFTs are subjected to different heating frequencies ω H The temperature rise ΔT under different heating frequencies 实验 (ω H ) is input into the theoretical calculation thermal model and ΔT 理论 (ω H ) is fitted to obtain the basic information of phonon scattering.

[0057] Reconstruct the phonon spectrum part, such as Figure 1 As shown in (c), the basic information of phonon scattering obtained by fitting is substituted into the BvK model to reconstruct the phonon MFT spectrum. This application explores the phonon spectrum from the time scale, that is, different mean free time τ ω The thermal conductivity of phonons. The following is a detailed introduction to the relevant definitions:

[0058] The intrinsic thermal conductivity of the material is:

[0059]

[0060] Where: k ω is the spectral function of the phonon frequency ω, specifically

[0061]

[0062] Where: C ω is the specific heat per unit volume of phonons with a phonon frequency of ω, v ω is the phonon group velocity, τ ω is the mean free time, and is used to intuitively represent different mean free times as τ ω The thermal conductivity of phonons, by substituting the integral variables, we can get:

[0063]

[0064] The phonon MFT spectrum function K τ for

[0065]

[0066] Where: K τ The mean free time is τ ω The intrinsic thermal conductivity k of the phonon pair bulk The corresponding cumulative function of thermal conductivity is

[0067]

[0068] α(τ α ) represents the intrinsic thermal conductivity k of the material bulk α% is determined by τ ω <τ α For example, at room temperature, the k bulk 50% of τ ω <τ α (≈0.1 ps) of phonons; 90% by τ ω <τ α (≈2.5 picoseconds) of phonons.

[0069] Formulas (1)-(5) are all intrinsic properties of the material, but when the external thermal excitation frequency ω H When it is increased to a certain value, different mean free time τ ω Phonons will be suppressed to varying degrees, and the mean free time will decrease by τ ω becomes τ eff. , at this time the phonon frequency spectrum function k ω for

[0070]

[0071] Phonon MFT spectrum function K τ for

[0072]

[0073] The suppression function S(t c ,L c ,ε ω )=τ eff / τ v ,S(t c ,L c ,∈ ω ) is about the characteristic time (heating frequency ω H ), characteristic size (heater or sample size), heating surface emissivity ε ω To simplify the problem, the traditional approach controls a single inhibitory factor, that is, DC heating or regulating the heater size and sample size of different sizes under low heating frequency; this application uses a large-size heater and body material to regulate the heating frequency ω H , so the suppression function S(ω H τ ω ,ε ω ) is only the heating frequency ω H and the heating surface emissivity ∈ ω At room temperature, the MFT range of phonons, which are mainly involved in heat conduction in semiconductor base material silicon, is roughly 10 -3 –10 3 Picoseconds, when the heating frequency ω H Below the MHz level, the phonons that mainly participate in heat conduction are not suppressed, and the experimentally measured equivalent thermal conductivity of the material is equal to the intrinsic bulk value of the material; when the heating frequency ω H When the frequency reaches the megahertz level, the material's equivalent thermal conductivity will decrease due to the non-local and non-equilibrium properties of the ballistic transport process. As the heating frequency further increases, the material's equivalent thermal conductivity will be 1-2 orders of magnitude smaller than the bulk value.

[0074] like Figure 2 As shown in the figure, the eigenspectrum of the phonon thermal conductivity of silicon at room temperature and its cumulative function are H =10 11 rad / s decreases after inhibition. Figure 2 (a) is the eigenvalue of the phonon frequency spectrum (Formula 2) and the heating frequency ω H Equivalent value after suppression (Formula 6); Figure 2 (b) is the eigenvalue of the phonon MFT spectrum (Formula 4) and the heating frequency ω H Equivalent value after suppression (Formula 7); Figure 2 (c) is the eigenvalue of the cumulative function of the phonon thermal conductivity (Formula 4 and Formula 5) and the heating frequency ω H Equivalent value after suppression (Formula 4 and Formula 7); Figure 2 (d) is the equivalent thermal conductivity with heating frequency ω H changes.

[0075] The basic idea of ​​reconstructing the phonon spectrum function in this application is to adjust different characteristic times (AC heating frequency) to change the equivalent thermal conductivity and temperature rise of the sample. c ,L c ,∈ ω )=τ eff / τ ω It is the key to obtain the phonon spectrum. The traditional approach is to numerically solve the experimental model based on the Boltzmann transport equation in theory, which requires a lot of computing resources and time; in the experiment, the thermal model based on the diffusion transport mechanism is used to fit the experimental data, which causes the problem of inconsistency in fitting the ballistic experimental data with the diffusion model. In order to solve the above two problems, this application solves the Boltzmann transport equation by an approximate method to obtain the analytical solution of temperature, heat flow, thermal conductivity and its inhibition function, and directly uses the analytical solution to fit the experimental data and extract the phonon spectrum information from it. The principle and details of establishing the thermal model are as follows:

[0076] The Boltzmann transport equation based on the relaxation time assumption expresses the phonon radiation intensity:

[0077]

[0078] ω is the phonon frequency, v ω is the phonon group velocity, is the unit direction vector of velocity, τ ω is the mean free time of phonons, I ω is the phonon radiation intensity:

[0079]

[0080] in is the reduced Planck constant, D(ω) is the phonon state density function, f ω is the phonon distribution function, For all directions I at a specific location ω The average radiation intensity. Formula (8) is a 7-dimensional integral-differential equation, which is difficult to solve directly analytically. This application approximates the phonon radiation intensity based on the first-order spherical harmonic function approximation as:

[0081]

[0082] The net heat flow Phonon incident radiation energy G ω =∫ 4π I ω dΩ, Ω is the solid angle of phonon radiation. Temperature and phonon incident radiation energy G ω Related:

[0083]

[0084] Substituting formula (10) into formula (8), we can directly multiply formula (8) by the solid angle and the velocity v ω The unit direction vector Then integrate the solid angle to get the net heat flow q net,ω and the incident radiation energy of the phonon G ω Coupled equations:

[0085]

[0086] Decouple equations (12) and (13):

[0087]

[0088] This application considers the periodic heating situation, then the incident phonon radiation energy and Substituting into equations (14) and (15), we can obtain the incident radiation energy G of the phonon: ω The governing equations are:

[0089]

[0090] and the net heat flow q net,ω and the incident radiation energy of the phonon G ω The relationship is as follows

[0091]

[0092] where α AC,ω =k AC,ω / C ω is the thermal diffusion coefficient of phonons with frequency ω, and in this application, the AC thermal conductivity k is defined as AC,ω (AC thermal conductivity):

[0093]

[0094] By introducing the AC thermal conductivity k AC,ω , the governing equation (16) and the traditional heat diffusion equation The form is the same, so the two have the same general solution form. The difference is that the bulk thermal diffusion coefficient α = k / C and the bulk thermal conductivity k are replaced by the AC thermal diffusion coefficient α AC,ω and AC thermal conductivity k AC,ω .

[0095] Considering the phonon non-equilibrium and non-local effects, the energy conservation equation at the boundary can be obtained to obtain the slip temperature boundary condition:

[0096]

[0097] And the heat flow boundary conditions:

[0098]

[0099] in, is the phonon radiation energy at the boundary, which is directly related to the boundary temperature, ε ω is the radiation coefficient, is the unit normal vector at the boundary.

[0100] like Figure 3 As shown, the theoretical framework of this application is summarized as follows: based on the diffusion control equation shown in formula (16), the solution of the existing diffusion equation is mapped to a ballistic solution to obtain a general solution form containing undetermined coefficients; the modified slip boundary conditions shown in formulas (19) and (20) are substituted to determine the constants of the solution; the equilibrium temperature T is obtained by solving formula (11) eq , using equation (17) to solve the net heat flow q net , and the boundary emission temperature T is obtained by solving equation (19) bdy ; Finally, the equivalent thermal conductivity k is obtained based on the heat flow and temperature. eff And the corresponding suppression function S.

[0101] The equivalent thermal conductivity and suppression function are related to the experimental conditions. According to the experimental sample size, heater size (length l and half width b), input excitation heat flux amplitude and frequency ω H =2ω I (with heating current amplitude I 1ω and frequency ω I Generally speaking, we will pre-estimate the required experimental conditions theoretically. For example, the MFT range of phonons, which are mainly involved in heat conduction in silicon, is about 10 -3 –10 3 Picoseconds, when the heating frequency ω H Below the MHz level, the suppression function is 1, the phonons mainly involved in heat conduction are not suppressed, and the experimentally measured equivalent thermal conductivity of the material is equal to the intrinsic bulk value of the material; when the heating frequency ω H When it reaches the megahertz level, limited by the non-local and non-equilibrium properties of the ballistic transport process, the inhibition function is less than 1, and the equivalent thermal conductivity of the material will decrease. As the heating frequency is further increased, the equivalent thermal conductivity of the material will be 1-2 orders of magnitude smaller than the bulk value.

[0102] Setting of heating current frequency: The approximate range of phonon MFT in the sample should be considered, which generally needs to reach above MHz.

[0103] The setting of the heating current amplitude needs to be considered in combination with the heater resistance and the thermal conductivity of the sample to ensure that the temperature rise generated can be detected after being converted into an electrical signal. Generally speaking, the temperature rise needs to be 10-100mK. At room temperature, for a bulk silicon sample with a thermal conductivity of 150W / mK, the heater resistance is 50-100 ohms, and the current amplitude is approximately 10mA. As the thermal conductivity decreases, the thermal resistance increases, and the current amplitude can be reduced.

[0104] The theoretical framework of the present application greatly simplifies the difficulty of solving the problem, reduces the requirements for computing resources and time, and can directly fit the experimentally measured temperature rise to the analytical solution, reconstructing the sample's intrinsic phonon MFT spectrum in a self-consistent, accurate and efficient manner.

[0105] like Figure 4 As shown, taking the classic 3ω method as an example of the experimental test method, the present application is used to reconstruct the intrinsic phonon MFT spectrum of silicon at room temperature. The workflow is as follows:

[0106] Experimental part, such as Figure 4 As shown in (a), in the first step, the "four-wire method" measures the resistance to be R e The “resistance-temperature characteristic” of the heater, that is, the temperature coefficient of resistance (TCR) of the heater; the second step is to pass an alternating current I of a certain magnitude and frequency through the current source to the micro heater. 1ω , heating frequency ω H The AC Joule heat flows into the silicon sample to be tested through the micro heater, and the Joule heat heating power is In the third step, the phonons of different MFTs in the silicon sample respond differently to thermal excitation. The average temperature of the microheater on the sample surface is related to the intrinsic phonon mean free time (MFT) spectrum of the silicon sample to be tested and the phonon penetration coefficient ε between the heater and the sample. In the fourth step, the phase-locked amplifier measures the triple frequency voltage V at both ends of the heater. 3ω In the basic principle of the 3 omega method, the temperature of the heater can be obtained by the heater resistance R e , drive current I 1ω , the micro heater resistance temperature coefficient TCR and the triple frequency voltage V 3ω Calculation of isoelectric signals

[0107]

[0108] The temperature signal is a complex number with both real and imaginary parts. The lock-in amplifier detects that the In-phase signal corresponds to the real part of the temperature rise, and the Out-phase signal corresponds to the imaginary part of the temperature rise. The present application obtains the phase of the temperature response based on the In-phase signal (real part of the temperature rise) and the Out-phase signal (imaginary part of the temperature rise). The physical meaning is the phase lag of temperature response relative to the excitation heat flow.

[0109] Theoretical part, such as Figure 4 As shown in (b), refer to Figure 3 The theoretical framework flow chart of this application shows that the control equation in the two-dimensional Cartesian coordinate system is:

[0110]

[0111] The modified boundary conditions are

[0112]

[0113] Among them, the radiation coefficient ε ω = is equal to the penetration rate of phonons between the heater and the sample, and H is the Heaviside step function. To simplify the problem, the penetration coefficient of phonons between the heater and the sample is a constant ε 加热器-样品 , does not change with the phonon frequency. Solving the above equation, the analytical expression for the theoretical temperature change of the sample is:

[0114]

[0115] Where: P0 is the amplitude of AC heat flow excitation, l is the length of the micro heater, C ω is the specific heat per unit volume of phonons with a phonon frequency of ω, v ω is the phonon group velocity, ε is the phonon penetration coefficient between the heater and the sample, λ is the Fourier cosine inverse transform integral variable, b is the half-width of the microheater, is the first intermediate parameter, H is the Heaviside step function, and t is time.

[0116] The amplitude of the AC heat flux excitation is:

[0117]

[0118] Where: I 1ω is the amplitude of the heating current, R e is the resistance of the heater.

[0119] i represents an imaginary unit, and λ represents an inverse Fourier cosine transform integral variable.

[0120] The theoretical phase lag can be obtained from the above formula: Contains the sample's intrinsic phonon mean free time (MFT) spectrum and the phonon penetration coefficient between the heater and the sample.

[0121] Step S3 includes:

[0122] Step S3-1: obtaining the real part and the imaginary part of the theoretical temperature change expression, and obtaining the theoretical phase lag based on the real part and the imaginary part of the theoretical temperature change expression;

[0123] Step S3-2: Substituting the measured temperature change of the sample, the amplitude and frequency of the AC heat flow excitation, and the size of the heater, the real part and the imaginary part of the measured temperature change expression are obtained, and the measured phase lag is obtained based on the real part and the imaginary part of the measured temperature change expression;

[0124] Step S3-3: fitting the measured phase lag to the theoretical phase lag to obtain the phonon scattering coefficient and the contribution of phonons of each mean free path to the thermal conductivity;

[0125] Step S3-4: Obtain a reconstructed phonon mean free time spectrum based on the phonon scattering coefficient and the contribution of phonons of each mean free path to the thermal conductivity.

[0126] like Figure 4 As shown in (c), based on the Bvk model, the experimentally measured phase lag φ 实验 Direct fit to the theoretical hysteresis φ 理论 , and obtain the phonon scattering coefficient information. Figure 4 d shows the phonon scattering coefficient obtained based on the fitting and the formula Reconstruct the phonon MFT spectrum function, and change the heater material to obtain the phonon penetration coefficient ε between different heaters and silicon samples 加热器-样品 .

[0127] For the phonon scattering coefficient, in semiconductor materials, the phonon mean free time τ ω Affected by phonon scattering, phonon scattering is divided into three categories: phonon-phonon scattering (umklapp), phonon-impurity scattering (impurity), and boundary scattering (boundary). According to Matthiessen's law, the three scattering modes and the phonon mean free time τ are established. ω Relationship: To simplify the problem and control a single inhibitory factor, starting from the time scale, that is, using a large-sized heater and a material without impurities and defects, the heating frequency ω is adjusted. H , so the phonon-impurity scattering (impurity) and boundary scattering (boundary) can be ignored, and the phonon mean free time τ ω Limited only by phonon-phonon scattering (umklapp), the expression is: where D and n are the scattering coefficients to be determined. ω Substitute the analytical expression of the theoretical temperature change of the sample ΔT 理论 , D and n are determined by fitting with the experimental temperature rise.

[0128] If the above functions are implemented in the form of software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, or the part of the technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium, including several instructions for a computer device (which can be a personal computer, a server, or a network device, etc.) to perform all or part of the steps of the method described in each embodiment of the present invention. The aforementioned storage medium includes: U disk, mobile hard disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), disk or optical disk, etc., which can store program code.

Claims

1. A method for reconstructing the mean free time spectrum of phonons in semiconductor materials, characterized in that: include: Step S1: Get different heating frequencies ω through experimental testing H The response temperature rise of the sample under the AC heat flow excitation; Step S2: Solving the Boltzmann transport equation according to the first-order spherical harmonic function approximation to obtain a diffusion equation, and combining it with the modified slip boundary condition to obtain an analytical expression for the theoretical temperature change of the sample, wherein in the analytical expression, the theoretical temperature change of the sample is related to the amplitude and frequency of the AC heat flow excitation, the size of the heater, the mean free time spectrum of the intrinsic phonons of the sample, and the phonon penetration coefficient between the heater and the sample; Step S3: After substituting the measured temperature change of the sample, the amplitude and frequency of the AC heat flux excitation, and the size of the heater, the experimentally measured response temperature rise change and the theoretical temperature change are fitted to obtain the sample's intrinsic phonon mean free time spectrum and the phonon penetration coefficient between the heater and the sample to complete the reconstruction of the phonon mean free time spectrum.

2. The method for reconstructing the mean free time spectrum of phonons in semiconductor materials according to claim 1, characterized in that: The analytical expression for the theoretical temperature change of the sample is: Where: P0 is the amplitude of AC heat flow excitation, l is the length of the micro heater, C ω is the specific heat per unit volume of phonons with a phonon frequency of ω, v ω is the phonon group velocity, ε is the phonon penetration coefficient between the heater and the sample, λ is the Fourier cosine inverse transform integral variable, b is the half-width of the microheater, is the first intermediate parameter, H is the Heaviside step function, and t is time.

3. The method for reconstructing the mean free time spectrum of phonons in semiconductor materials according to claim 2, characterized in that: The amplitude of the AC heat flux excitation is: Where: I 1ω is the amplitude of the heating current, R e is the resistance of the heater.

4. The method for reconstructing the mean free time spectrum of phonons in semiconductor materials according to claim 2, characterized in that: The first intermediate parameter is: Where: k AC,ω is the AC thermal conductivity, is the second intermediate parameter.

5. The method for reconstructing the mean free time spectrum of phonons in semiconductor materials according to claim 4, characterized in that: The second intermediate parameter is: Where: H is the heating frequency, α AC,ω is the AC thermal diffusion coefficient; The AC thermal conductivity is: Where: τ ω is the average free time.

6. The method for reconstructing the mean free time spectrum of phonons in semiconductor materials according to claim 4, characterized in that: The modified slip boundary condition is: Where: q net,bdy (x, y = 0) is the input excitation heat flux distribution at the upper surface boundary of the sample, G ω (y=∞) is the incident radiation energy of phonons at the bottom boundary of the sample, ∈ ω is the penetration coefficient of phonons with a phonon frequency of ω between the heater and the sample, is the unit normal vector at the boundary.

7. The method for reconstructing the mean free time spectrum of phonons in semiconductor materials according to claim 4, characterized in that: The step S3 comprises: Step S3-1: obtaining the real part and the imaginary part of the theoretical temperature change expression, and obtaining the theoretical phase lag based on the real part and the imaginary part of the theoretical temperature change expression; Step S3-2: Substituting the measured temperature change of the sample, the amplitude and frequency of the AC heat flow excitation, and the size of the heater, the real part and the imaginary part of the measured temperature change expression are obtained, and the measured phase lag is obtained based on the real part and the imaginary part of the measured temperature change expression; Step S3-3: fitting the measured phase lag to the theoretical phase lag to obtain the phonon scattering coefficient and the contribution of phonons of each mean free path to the thermal conductivity; Step S3-4: Obtain a reconstructed phonon mean free time spectrum based on the phonon scattering coefficient and the contribution of phonons of each mean free path to the thermal conductivity.

8. The method for reconstructing the mean free time spectrum of phonons in semiconductor materials according to claim 7, characterized in that: The contribution of the mean free path of phonons to the thermal conductivity is: Where: K τ is the phonon MFT spectrum function, indicating that the mean free time is τ ω The thermal conductivity of phonons.

9. A device for reconstructing the mean free time spectrum of phonons in semiconductor materials, comprising a memory, a processor, and a program stored in the memory, characterized in that: When the processor executes the program, the method according to any one of claims 1 to 8 is implemented.

10. A storage medium having a program stored thereon, characterized in that: When the program is executed, the method according to any one of claims 1 to 8 is implemented.

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