CN113987841B - Method for solving phonon heat transport at interface and storage medium - Google Patents
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Abstract
The invention relates to a method and a storage medium for solving phonon heat transport at an interface, wherein the method comprises the following steps: initializing set parameters and system parameters; acquiring phonon deviation intensity based on the system parameters; iteratively updating the phonon deviation intensity based on a linear Boltzmann equation and a phonon interface condition, and obtaining a corresponding temperature, a pseudo temperature and a heat flow based on any one of the phonon deviation intensities; obtaining a difference value between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and defining the difference value as a relative error; and comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity. The method can directly, accurately and efficiently solve the heat transport of the phonons at the interface in the semiconductor, and can ensure the accuracy of the solved result to the maximum extent.
Description
Technical Field
The invention relates to the technical field of heat transfer, in particular to a method for solving phonon heat transport at an interface and a storage medium.
Background
The problem of heat dissipation in electronic devices has become a significant obstacle to their further development. The research on the phonon heat transport mechanism in the semiconductor can provide effective guidance for the heat dissipation optimization design of electronic devices, and the research method mainly comprises theoretical calculation, experimental research and numerical simulation. The discrete coordinate method is a numerical method based on direct solving of the phonon boltzmann equation, and is an important numerical tool for researching phonon heat transport in semiconductors. The discrete coordinate method has obvious advantages for simulating the mesoscale system with simple geometric shape, the algorithm is simple to realize, and the simulation precision is high. However, the numerical algorithm framework of the phonon discrete coordinate method is still imperfect.
In the prior art, the phonon discrete coordinate method considering the actual dispersion relation is mainly divided into two categories: and a numerical algorithm based on a linear equation under small temperature difference and a nonlinear equation under any temperature difference. The two kinds of numerical algorithms can further consider the steady state situation and the transient state situation respectively, and correspond to the space domain algorithm and the time-space domain algorithm. Numerical algorithms based on linear boltzmann equations are simpler and more efficient than numerical algorithms based on non-linear boltzmann equations. The method is different from the former method in that the information exchange mechanism of phonons at the interface in materials at two sides of the interface is required to be considered in the latter method, so that the physical process is more complicated. At present, for a phonon discrete coordinate method based on a linearized equation, a discrete coordinate method space domain and a time-space domain algorithm framework without considering an interface system and a discrete coordinate method space domain algorithm framework with considering the interface system are established, but the existing discrete coordinate method time-space domain algorithm framework with considering the interface system still needs to be further improved so as to meet the more and more urgent numerical simulation requirement of phonon heat transport research in semiconductors.
Disclosure of Invention
In order to solve the technical problems, the invention provides a method and a storage medium for solving the phonon heat transport at the interface, which can directly, accurately and efficiently solve the phonon heat transport at the interface in a semiconductor and furthest ensure the accuracy of a solving result.
In order to achieve the above object, the present application proposes a first technical solution:
a method of resolving phonon thermal transport at an interface, the method comprising the steps of: step S1, initializing setting parameters and system parameters; step S2, acquiring phonon deviation intensity based on the system parameters; step S3, iteratively updating the phonon deviation intensity based on a linear Boltzmann equation and a phonon interface condition, and obtaining corresponding temperature, pseudo-temperature and heat flow based on any one phonon deviation intensity; step S4, obtaining the difference value between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and defining the difference value as a relative error; and step S5, comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In one embodiment, the method further comprises: step S6, obtaining the calculation duration of the iterative update of the phonon deviation intensity, and defining the calculation duration as a first calculation duration; step S7, if the first calculated time length is less than the preset total calculated time length, continuing to execute the steps S3, S4, S5, S6 and S7; otherwise, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In one embodiment, the setting parameters include: space step length, system scale, time step length, total calculation time length and convergence error; the system parameters include: temperature and pseudo-temperature; the phonon bias intensities include: a first deviation intensity obtained based on the phonon intensity and a second deviation intensity obtained based on the phonon pseudo-equilibrium intensity.
In one embodiment, the step S2 specifically includes: step S21, setting the initial value of the temperature as T0And acquiring the phonon intensity as follows based on the initial value of the temperature:
in the formula (I), the compound is shown in the specification,
it is the intensity of the phonon that is,
which is the velocity of the phonon group,
in order to approximate the planck constant,
in order to be the angular frequency of the phonons,pin order to be a phonon branch,
is the distribution of the phonons and the phonons,Dis the density of the phonon states,
representing a phonon branch aspAnd the phonon angular frequency is
Phonon state density of (a); step S22, based on the phonon intensity, obtaining the phonon reference balance intensity as follows:
in the formula (I), the compound is shown in the specification,
for the phonon reference to the equilibrium intensity,
to take into account a constant type reference temperature
The bose-einstein distribution of (a); step S23, obtaining a first deviation intensity based on the phonon intensity and the phonon reference balance intensity:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
it is the intensity of the phonon that is,
balance intensity is referred to as phonon; step S24, setting the initial value of the pseudo temperature and the initial value of the balance value to be equal to each other and to be T0And acquiring the phonon pseudo-equilibrium intensity based on the initial value of the pseudo-temperature as follows:
in the formula (I), the compound is shown in the specification,
for the pseudo-equilibrium intensity of the phonons,
is to take into account the initial pseudo-temperature
The bose-einstein distribution of (a); step S25, obtaining a second deviation intensity based on the phonon intensity and the phonon pseudo-equilibrium intensity:
in the formula (I), the compound is shown in the specification,
is the second deviation intensity.
In one embodiment, the linear boltzmann equation is:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
in order to be the second deviation strength,
in order to be the phonon relaxation time,
the angular frequency of the phonons is represented,pthe number of phonon branches is represented,
which represents the velocity of the phonon group,
representing a phonon branch aspAnd the phonon angular frequency is
(ii) a phonon group velocity of; the interface conditions are as follows:
wherein A, B respectively represent two materials,
is the included angle between the normal vector of the interface and the group velocity, the positive direction of the normal direction of the interface points to the material B from the material A,
and
spectral interface transmission coefficients from material a to material B and from material B to material a respectively,
is the first off-set strength of material B,
is the first deflection strength of material a.
In one embodiment, the obtaining the corresponding temperature, pseudo-temperature, and heat flow based on any phonon bias intensity specifically includes: step S31, obtaining a linear expression of temperature based on the basic definitions of the phonon deviation intensity and temperature:
in the formula (I), the compound is shown in the specification,
，
to take into account the temperatureTThe glass-einstein distribution at the time,
is a constant type of reference temperature, and,
in the form of a polar coordinate, the position of the lens,
in order to be the azimuth angle,
representing phonon branchespThe maximum phonon angular frequency of (a) is,
which is the velocity of the phonon group,
in order to be the first deviation strength,
the volume heat capacity of the frequency spectrum, namely the volume heat capacity of each phonon branch and a unit phonon angular frequency interval;
wherein the temperature is defined essentially as:
step S32, obtaining a linear expression of the pseudo temperature based on the phonon deviation intensity and the basic definition of the pseudo temperature:
in the formula (I), the compound is shown in the specification,
in order to be the pseudo-temperature,
is the phonon relaxation time;
wherein the pseudo-temperature is substantially defined as:
step S33, acquiring a linear expression of the obtained heat flow based on the phonon deviation intensity:
in the formula (I), the compound is shown in the specification,qis the heat flow.
In one embodiment, the step S5 specifically includes: step S51, obtaining a relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:
in the formula (I), the compound is shown in the specification,
the relative error is indicated in the form of,nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,Tis the temperature;
and step S52, if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In one embodiment, the step S5 specifically includes:
step S53, obtaining a difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and defining the difference as a relative error:
in the formula (I), the compound is shown in the specification,
is a relative error;nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,qis a heat flow;
and step S54, comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In order to achieve the above object, the present application proposes a second technical solution:
a computer-readable storage medium storing a program which, when executed by a processor, causes the processor to perform the steps of the method.
Compared with the prior art, the technical scheme of the invention has the following advantages:
the method and the storage medium for solving the phonon heat transport at the interface can directly, accurately and efficiently solve the phonon heat transport at the interface in the semiconductor, and can ensure the accuracy of a solving result to the maximum extent.
Drawings
In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings needed to be used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without creative efforts.
FIG. 1 is a flow chart of a method of the present invention for solving for phonon heat transport at an interface;
FIG. 2 is a flow chart of a method of the present invention for solving for phonon heat transport at an interface;
FIG. 3 is a schematic structural diagram of a double-layer film according to a third embodiment of the present invention;
FIG. 4 is a graph showing the experimental results of temperature evolution of the double-layer film according to the third embodiment of the present invention;
fig. 5 is a diagram illustrating the experimental results of the thermal evolution of the double-layer film according to the third embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The first embodiment is as follows:
referring to fig. 1, fig. 1 is a flowchart of a method according to a first embodiment.
The method of the embodiment comprises the following steps:
step S1, initializing setting parameters; the setting parameters comprise a space step length, a system scale, a time step length, total calculation time length and a convergence error; the system scale is the spatial scale of the system, such as: the length, width and height of the system. Since it is only concerned with the one-dimensional space problem, that is, the present application only considers the variation of the system parameters in the thickness direction of the experimental material, that is, if only the variation of the system in the length direction of the experimental material is considered, it is assumed that the width and height of the experimental material are infinite. The total time step is the total simulated time length when calculating the phonon heat transport of the experimental material at the interface, and the time step is the sampling point of one of the total time steps, that is, the total time step is composed of several time periodsAnd (4) the step size is reduced. Such as: in a simulation experiment, experimental simulation experiment materials from 0 to 1000 picoseconds: (ps) Internal temperature change, of which 1000psThe time length of (a) is the total time step. According to the actual experimental needs, 1000 ispsDivided into 1000 sampling points, i.e. 1psSampling once, then 1psBecomes the first sample point, which is the time step. Wherein the system parameters include phonon intensity and phonon pseudo-equilibrium intensity. It should be understood that, in order to avoid the divergence of the calculation results and increase the convergence speed, one skilled in the art should refer to the average values of the phonon mean free path and the relaxation time when setting the space step size and the time step size, and should make the space step size and the time step size as small as possible than the average values of the phonon mean free path and the relaxation time.
Step S2, setting initial value of phonon intensity and pseudo-equilibrium intensity value of phonon; this step S2 specifically includes: obtaining corresponding phonon intensity, phonon reference temperature and first deviation intensity based on the initial value of the temperature; and obtaining the corresponding phonon pseudo-equilibrium intensity and second deviation intensity based on the initial value of the pseudo-temperature.
In a specific embodiment, obtaining the corresponding phonon intensity, phonon reference temperature and first deviation intensity based on the initial value of the temperature specifically comprises the following steps: step S21, setting the initial value of the temperature as T0At a known temperature, the initial value is T0In the case of (2), the phonon intensity is solved based on the following equation:
in the formula (I), the compound is shown in the specification,
it is the intensity of the phonon that is,
which is the velocity of the phonon group,
in order to approximate the planck constant,
in order to be the angular frequency of the phonons,pin order to be a phonon branch,
is the distribution of the phonons and the phonons,Dis the density of the phonon states,
representing a phonon branch aspAnd the phonon angular frequency is
Phonon state density of (a).
Step S22, based on the phonon intensity, obtaining the phonon reference balance intensity as follows:
in the formula (I), the compound is shown in the specification,
for the phonon reference to the equilibrium intensity,
to take into account a constant type reference temperature
The bose-einstein distribution of (a).
Step S23, obtaining a first deviation intensity based on the phonon intensity and the phonon reference balance intensity:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
it is the intensity of the phonon that is,
the equilibrium intensity is referred to as the phonon.
In a specific embodiment, obtaining the corresponding phonon pseudo-equilibrium intensity and the second deviation intensity based on the initial value of the pseudo-temperature specifically includes:
step S24, setting the initial value of the pseudo temperature and the initial value of the balance value to be equal to each other and to be T0I.e. it is assumed here that both the phonon temperature and the phonon pseudo-temperature value are T0And acquiring the phonon pseudo-equilibrium intensity based on the initial value of the pseudo-temperature as follows:
in the formula (I), the compound is shown in the specification,
for the pseudo-equilibrium intensity of the phonons,
is to take into account the initial pseudo-temperature
The bose-einstein distribution of (a);
step S25, obtaining a second deviation intensity based on the phonon intensity and the phonon pseudo-equilibrium intensity:
in the formula (I), the compound is shown in the specification,
is the second deviation intensity.
Step S3, obtaining a first phonon intensity based on the linear boltzmann equation. That is, the phonon intensity obtained in step S21 is iteratively updated based on the linear boltzmann equation, and the obtained phonon intensity after iterative updating is defined as the first phonon intensity.
In a specific embodiment, the obtaining the first phonon intensity based on the boltzmann equation specifically includes: the linear boltzmann equation is:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
in order to be the second deviation strength,
in order to be the phonon relaxation time,
the angular frequency of the phonons is represented,pthe number of phonon branches is represented,
which represents the velocity of the phonon group,
representing a phonon branch aspAnd the phonon angular frequency is
The velocity of the phonon group. It is to be understood that the application is based on the characteristics of the space-time discrete coordinate method, and the phonon distribution is not only dependent on the space coordinate, but also dependent on the timeAnd (4) coordinates. Therefore, iterative updating is carried out on the obtained phonon intensity by adopting a linear Boltzmann equation. Specifically, the differential format is adopted for the time and space differentiation, so that the change rule of the system temperature and the heat flow along the space coordinate can be obtained, and the evolution rule of the macroscopic information changing along with time can also be obtained. Since the total computation time is divided into several time steps in the present application, and the thermal transport of phonons at the interface is solved based on any time step, the mutual exchange of phonon information at the interface needs to be considered, that is, in the actual experimental process, the interface conditions need to be considered, where the interface conditions are:
wherein A, B respectively represent two materials,
is the included angle between the normal vector of the interface and the group velocity, the positive direction of the normal direction of the interface points to the material B from the material A,
and
spectral interface transmission coefficients from material a to material B and from material B to material a respectively,
is the first off-set strength of material B,
is the first deflection strength of material a. It should be understood that two adjacent materials are exemplified, and in practical situations, even if the same material has a plurality of materials different from the adjacent materials, one skilled in the art can divide any two adjacent materials intoA set takes into account the interface conditions.
Step S4, obtaining a temperature, a pseudo temperature, and a heat flow based on the first phonon intensity. The method specifically comprises the following steps:
step S31, obtaining a linear expression of temperature based on the basic definitions of the phonon deviation intensity and temperature:
in the formula (I), the compound is shown in the specification,
，
to take into account the temperatureTThe glass-einstein distribution at the time,
is a constant type of reference temperature, and,
in the form of a polar coordinate, the position of the lens,
in order to be the azimuth angle,
representing phonon branchespThe maximum phonon angular frequency of (a) is,
which is the velocity of the phonon group,
in order to be the first deviation strength,
is the spectral volumetric heat capacity, i.e. the volumetric heat per phonon branch, per phonon angular frequency intervalC, holding;
wherein the temperature is defined essentially as:
step S32, obtaining a linear expression of the pseudo temperature based on the phonon deviation intensity and the basic definition of the pseudo temperature:
in the formula, in the formula:
in order to be the pseudo-temperature,
is the phonon relaxation time;
wherein the pseudo-temperature is substantially defined as:
step S33, acquiring a linear expression of the obtained heat flow based on the phonon deviation intensity:
in the formula (I), the compound is shown in the specification,qis the heat flow.
Obtaining a difference value between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and defining the difference value as a relative error;
and step S5, acquiring the relative error of the two phases of the approximate temperature. The method specifically comprises the following steps:
step S51, obtaining a relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:
in the formula (I), the compound is shown in the specification,
the relative error is indicated in the form of,nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,Tis the temperature;
and step S52, if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity, and continuously judging whether the calculation time is greater than or equal to the final evolution time. Namely, whether the current calculation time length exceeds the preset total calculation time length is judged. It should be understood that, here, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity is to output the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity at the current time step. Because the total calculation time length is composed of a plurality of time step lengths, the purpose of judging whether the current calculation time length exceeds the preset total calculation time length is as follows: and if the current calculation time does not exceed the preset total calculation time, calculating the temperature, the pseudo temperature and the heat flow of the next time step.
In a specific embodiment, step S5 obtains the relative error of two near temperatures. The method specifically comprises the following steps:
step S53, obtaining a difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and defining the difference as a relative error:
in the formula (I), the compound is shown in the specification,
is a relative error;nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,qis a heat flow;
and step S54, comparing the relative error with a preset convergence error, if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity, and continuously judging whether the calculation time is greater than or equal to the final evolution time. Namely, whether the current calculation time length exceeds the preset total calculation time length is judged. It should be understood that, here, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity is to output the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity at the current time step. Because the total calculation time length is composed of a plurality of time step lengths, the purpose of judging whether the current calculation time length exceeds the preset total calculation time length is as follows: and if the current calculation time does not exceed the preset total calculation time, calculating the temperature, the pseudo temperature and the heat flow of the next time step. It should be understood that, in an actual application scenario, a person skilled in the art may select to obtain a difference between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity through steps S51 and S52, or obtain a difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity through steps S53 and S54, and compare the obtained difference with a preset convergence error to determine whether the result meets the requirement. Similarly, one skilled in the art can also choose to compare the difference between the current heat flow and the previous heat flow and the difference between the current temperature and the previous temperature with the preset convergence errors respectively set corresponding to the two in an experiment to ensure that the result meets the actual requirement.
In a specific embodiment, the step of continuously determining whether the calculation time is greater than or equal to the final evolution time specifically includes the following steps:
step S6, obtaining the calculation duration of the iterative update of the phonon deviation intensity, and defining the calculation duration as a first calculation duration;
step S7, if the first calculated time length is less than the preset total calculated time length, continuing to execute the steps S3, S4, S5, S6 and S7; otherwise, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity. It should be understood that the process described in the embodiment is a process for solving the phonon heat transport at the interface in one time step. If the first calculation time length is less than the preset total calculation time length, continuing to execute the steps S3, S4, S5, S6 and S7, namely solving the phonon heat transport at the phonon position of the next time step; otherwise, if the first calculation time length is greater than or equal to the preset total calculation time length, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity. Moreover, it should be understood that, if the first calculation duration is less than the preset total calculation duration, the steps S3, S4, S5, S6 and S7 are continuously performed, that is, the next iterative update of the first phonon intensity is performed, that is, the current first phonon intensity is substituted into the step S3, and the current first phonon intensity is iteratively updated to obtain the new first phonon intensity until the first calculation duration is greater than or equal to the preset total calculation duration. It should be understood that, in this embodiment, no specific value is given for any preset setting parameter and system parameter such as convergence error, total computation time, time step, etc., and those skilled in the art may reasonably select the setting parameter and the system parameter according to the actual situation, and only need to satisfy the relevant requirements in this embodiment.
Example two:
referring to fig. 2, fig. 2 is a flowchart of a method according to a second embodiment.
The method for solving the phonon heat transport at the interface comprises the following steps: step S1, initializing setting parameters and system parameters; step S2, acquiring phonon deviation intensity based on the system parameters; step S3, iteratively updating the phonon deviation intensity based on a linear Boltzmann equation and a phonon interface condition, and obtaining corresponding temperature, pseudo-temperature and heat flow based on any one phonon deviation intensity; step S4, obtaining the difference value between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and defining the difference value as a relative error; and step S5, comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In one embodiment, the method further comprises: step S6, obtaining the calculation duration of the iterative update of the phonon deviation intensity, and defining the calculation duration as a first calculation duration; step S7, if the first calculated time length is less than the preset total calculated time length, continuing to execute the steps S3, S4, S5, S6 and S7; otherwise, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In one embodiment, the setting parameters include: space step length, system scale, time step length, total calculation time length and convergence error; the system parameters include: temperature and pseudo-temperature; the phonon bias intensities include: a first deviation intensity obtained based on the phonon intensity and a second deviation intensity obtained based on the phonon pseudo-equilibrium intensity.
In an embodiment, the step S2 specifically includes: step S21, setting the initial value of the temperature as T0And acquiring the phonon intensity as follows based on the initial value of the temperature:
in the formula (I), the compound is shown in the specification,
it is the intensity of the phonon that is,
which is the velocity of the phonon group,
in order to approximate the planck constant,
in order to be the angular frequency of the phonons,pin order to be a phonon branch,
is the distribution of the phonons and the phonons,Dis the density of the phonon states,
representing a phonon branch aspAnd the phonon angular frequency is
Phonon state density of (a);
step S22, based on the phonon intensity, obtaining the phonon reference balance intensity as follows:
in the formula (I), the compound is shown in the specification,
for the phonon reference to the equilibrium intensity,
to take into account a constant type reference temperature
The bose-einstein distribution of (a);
step S23, obtaining a first deviation intensity based on the phonon intensity and the phonon reference balance intensity:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
it is the intensity of the phonon that is,
balance intensity is referred to as phonon;
step S24, setting the initial value of the pseudo temperature and the initial value of the balance value to be equal to each other and to be T0And acquiring the phonon pseudo-equilibrium intensity based on the initial value of the pseudo-temperature as follows:
in the formula (I), the compound is shown in the specification,
for the pseudo-equilibrium intensity of the phonons,
is to take into account the initial pseudo-temperature
The bose-einstein distribution of (a);
step S25, obtaining a second deviation intensity based on the phonon intensity and the phonon pseudo-equilibrium intensity:
in the formula (I), the compound is shown in the specification,
is the second deviation intensity.
In one embodiment, the linear boltzmann equation is:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
in order to be the second deviation strength,
in order to be the phonon relaxation time,
the angular frequency of the phonons is represented,pthe number of phonon branches is represented,
which represents the velocity of the phonon group,
representing a phonon branch aspAnd the phonon angular frequency is
(ii) a phonon group velocity of;
the interface conditions are as follows:
wherein A, B respectively represent two materials,
is the included angle between the normal vector of the interface and the group velocity, the positive direction of the normal direction of the interface points to the material B from the material A,
and
spectral interface transmission coefficients from material a to material B and from material B to material a respectively,
is the first off-set strength of material B,
is the first deflection strength of material a.
In one embodiment, the obtaining the corresponding temperature, pseudo-temperature, and heat flow based on any phonon bias intensity specifically includes: step S31, obtaining a linear expression of temperature based on the basic definitions of the phonon deviation intensity and temperature:
in the formula (I), the compound is shown in the specification,
，
to take into account the temperatureTThe glass-einstein distribution at the time,
is a constant type of reference temperature, and,
in the form of a polar coordinate, the position of the lens,
in order to be the azimuth angle,
representing phonon branchespThe maximum phonon angular frequency of (a) is,
which is the velocity of the phonon group,
in order to be the first deviation strength,
the volume heat capacity of the frequency spectrum, namely the volume heat capacity of each phonon branch and a unit phonon angular frequency interval;
wherein the temperature is defined essentially as:
step S32, obtaining a linear expression of the pseudo temperature based on the phonon deviation intensity and the basic definition of the pseudo temperature:
in the formula:
in order to be the pseudo-temperature,
is the phonon relaxation time;
wherein the pseudo-temperature is substantially defined as:
step S32, the basic definition of the pseudo temperature is:
step S33, acquiring a linear expression of the obtained heat flow based on the phonon deviation intensity:
in the formula (I), the compound is shown in the specification,qis the heat flow.
In an embodiment, the step S5 specifically includes: step S51, obtaining a relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:
in the formula (I), the compound is shown in the specification,
the relative error is indicated in the form of,nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,Tis the temperature; and step S52, if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In an embodiment, the step S5 specifically includes:
step S53, obtaining a difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and defining the difference as a relative error:
in the formula (I), the compound is shown in the specification,
is a relative error;nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,qis a heat flow; step S54, comparing the relative error with a preset convergence error, if the relative error is not larger than the convergence error, thenAnd outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
Example three:
in a specific embodiment, a two-layer film is taken as an example, and one-dimensional unsteady phonon thermal transport is solved. Specifically, the bilayer film was composed of germanium and silicon of equal thickness, with a film thickness of 120 nm. As shown in FIG. 3, isothermal boundaries are arranged at two ends of the thin film and are respectively fixed at 301K and 299K, the requirement of small temperature difference at two sides of the interface is met, the initial temperature inside the thin film is uniform at 300K, and the temperature and the heat flow gradually tend to be stable along with time evolution. The temperature evolution diagrams and the heat flow evolution diagrams of the double-layer film at a plurality of different time steps, which are obtained by solving based on the Monte Carlo method, are taken as reference, and the accuracy of the two methods is verified based on the traditional method for solving the phonon heat transport at the interface and the method for solving the phonon heat transport at the interface provided by the application.
In a specific embodiment, as shown in fig. 4, that is, based on the double-layer film shown in fig. 3, the temperature evolution diagrams of the double-layer film at a plurality of different time steps are obtained by solving based on the prior art, the method described in the present application, and the monte carlo method under the set parameters. The method specifically comprises the following steps: initialization space step size of 0.2 nm and time step size of 0.2 nmpsTotal calculation time of 0.6 ns, convergence error 1 x 10-8The initial value of the temperature is 300 kelvin. Wherein, the time step length is 6psA time step of 48psAnd a time step of 600psThe temperature evolution curve of the double-layer film obtained by the method, the traditional method for solving the phonon heat transport at the interface and the Monte Carlo method at any time step are included, and the temperature evolution curve of the double-layer film obtained by the three methods at any time step is displayed together to play a role of comparison. It can be seen from fig. 4 that the temperature profile calculated by the conventional method is much different from the correct value obtained by the monte carlo-based method, whereas the temperature profile obtained by the method described in the present application is different from the correct value obtained by the monte carlo-based methodThe difference of the correct values obtained by the TerCarlo method is small, so that the method for solving the phonon heat transport at the interface can ensure the accuracy and precision of the calculation result to the maximum extent.
In one embodiment, as shown in fig. 5, that is, based on the dual-layer film shown in fig. 3, the heat flow evolution diagrams of the dual-layer film at a plurality of different time steps are obtained by solving based on the prior art, the method described in the present application, and the monte carlo method under the set parameters. Wherein the initialization space step is 0.2 nm, and the time step is 0.2psTotal calculation time of 0.6 ns, convergence error 1 x 10-8The initial value of the temperature is 300 kelvin. Wherein, FIG. 5 includes a time step of 6psA time step of 48psAnd a time step of 600psThe heat flow evolution curve of the double-layer film obtained by the method, the traditional method for solving the phonon heat transport at the interface and the Monte Carlo method at any time step are included, and the heat flow evolution curve of the double-layer film obtained by the three methods at any time step is displayed together to play a role of comparison. As can be seen from fig. 5, the difference between the curve graph of the heat flow calculated by the conventional method and the correct value obtained by the monte carlo method is large, and the difference between the curve graph of the heat flow obtained by the method of the present application and the correct value obtained by the monte carlo method is small, which proves that the method for solving the phonon heat transport at the interface of the present application can ensure the accuracy and precision of the calculation result to the maximum extent. It is to be understood that the X-axis in both fig. 4 and 5 represents a dimensionless spatial coordinate, specifically the ratio between the spatial coordinate and the film thickness; the Y-axis of fig. 4 represents temperature in kelvin; the Y-axis of fig. 5 represents heat flow in watts per square meter.
Example four:
the present embodiment provides a computer-readable storage medium storing a program that, when executed by a processor, causes the processor to perform the steps of the method of the first and second embodiments.
In a specific embodiment, the program when executed by a processor implements the steps of:
step S1, initializing setting parameters and system parameters; step S2, acquiring phonon deviation intensity based on the system parameters; step S3, iteratively updating the phonon deviation intensity based on a linear Boltzmann equation and a phonon interface condition, and obtaining corresponding temperature, pseudo-temperature and heat flow based on any one phonon deviation intensity; step S4, obtaining the difference value between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and defining the difference value as a relative error; and step S5, comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity. Wherein the setting parameters include: space step length, system scale, time step length, total calculation time length and convergence error; the system parameters include: temperature and pseudo-temperature; the phonon bias intensities include: a first deviation intensity obtained based on the phonon intensity and a second deviation intensity obtained based on the phonon pseudo-equilibrium intensity.
In a specific embodiment, the program when executed by a processor implements the steps of:
step S6, obtaining the calculation duration of the iterative update of the phonon deviation intensity, and defining the calculation duration as a first calculation duration; step S7, if the first calculated time length is less than the preset total calculated time length, continuing to execute the steps S3, S4, S5, S6 and S7; otherwise, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In a specific embodiment, the program when executed by a processor implements the steps of:
step S21, setting the initial value of the temperature as T0And acquiring the phonon intensity as follows based on the initial value of the temperature:
in the formula (I), the compound is shown in the specification,
it is the intensity of the phonon that is,
which is the velocity of the phonon group,
in order to approximate the planck constant,
in order to be the angular frequency of the phonons,pin order to be a phonon branch,
is the distribution of the phonons and the phonons,Dis the density of the phonon states,
representing a phonon branch aspAnd the phonon angular frequency is
Phonon state density of (a);
step S22, based on the phonon intensity, obtaining the phonon reference balance intensity as follows:
in the formula (I), the compound is shown in the specification,
for the phonon reference to the equilibrium intensity,
to take into account a constant type reference temperature
The bose-einstein distribution of (a);
step S23, obtaining a first deviation intensity based on the phonon intensity and the phonon reference balance intensity:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
it is the intensity of the phonon that is,
balance intensity is referred to as phonon;
step S24, setting the initial value of the pseudo temperature and the initial value of the balance value to be equal to each other and to be T0And acquiring the phonon pseudo-equilibrium intensity based on the initial value of the pseudo-temperature as follows:
in the formula (I), the compound is shown in the specification,
for the pseudo-equilibrium intensity of the phonons,
is to take into account the initial pseudo-temperature
The bose-einstein distribution of (a);
step S25, obtaining a second deviation intensity based on the phonon intensity and the phonon pseudo-equilibrium intensity:
in the formula (I), the compound is shown in the specification,
is the second deviation intensity.
In a specific embodiment, the linear boltzmann equation is:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
in order to be the second deviation strength,
in order to be the phonon relaxation time,
the angular frequency of the phonons is represented,pthe number of phonon branches is represented,
which represents the velocity of the phonon group,
representing a phonon branch aspAnd the phonon angular frequency is
(ii) a phonon group velocity of;
the interface conditions are as follows:
wherein A, B respectively represent two materials,
is the included angle between the normal vector of the interface and the group velocity, the positive direction of the normal direction of the interface points to the material B from the material A,
and
spectral interface transmission coefficients from material a to material B and from material B to material a respectively,
is the first off-set strength of material B,
is the first deflection strength of material a.
In a specific embodiment, the program when executed by a processor implements the steps of:
step S31, obtaining a linear expression of temperature based on the basic definitions of the phonon deviation intensity and temperature:
in the formula (I), the compound is shown in the specification,
，
to take into account the temperatureTThe glass-einstein distribution at the time,
is a constant type of reference temperature, and,
in the form of a polar coordinate, the position of the lens,
in order to be the azimuth angle,
representing phonon branchespThe maximum phonon angular frequency of (a) is,
which is the velocity of the phonon group,
in order to be the first deviation strength,
the volume heat capacity of the frequency spectrum, namely the volume heat capacity of each phonon branch and a unit phonon angular frequency interval;
wherein the temperature is defined essentially as:
step S32, obtaining a linear expression of the pseudo temperature based on the phonon deviation intensity and the basic definition of the pseudo temperature:
in the formula:
in order to be the pseudo-temperature,
is the phonon relaxation time;
wherein the pseudo-temperature is substantially defined as:
step S33, acquiring a linear expression of the obtained heat flow based on the phonon deviation intensity:
in the formula (I), the compound is shown in the specification,qis the heat flow.
In a specific embodiment, the program when executed by a processor implements the steps of:
step S51, obtaining a relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:
in the formula (I), the compound is shown in the specification,
the relative error is indicated in the form of,nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,Tis the temperature;
and step S52, if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
In a specific embodiment, the program when executed by a processor implements the steps of:
step S53, obtaining a difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and defining the difference as a relative error:
in the formula (I), the compound is shown in the specification,
is a relative error;nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,qis a heat flow;
and step S54, comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
As will be appreciated by one of skill in the art, embodiments of the present invention may be provided as a method, system, or computer program product. Accordingly, embodiments of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, embodiments of the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
Embodiments of the present invention are described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
It is to be noted that the foregoing is only illustrative of the preferred embodiments of the present invention and the technical principles employed. It will be understood by those skilled in the art that the present invention is not limited to the particular embodiments described herein, but is capable of various obvious changes, rearrangements and substitutions as will now become apparent to those skilled in the art without departing from the scope of the invention. Therefore, although the present invention has been described in greater detail by the above embodiments, the present invention is not limited to the above embodiments, and may include other equivalent embodiments without departing from the spirit of the present invention, and the scope of the present invention is determined by the scope of the appended claims.
Claims (10)
1. A method for solving phonon heat transport at an interface is characterized in that: the method comprises the following steps:
step S1, initializing setting parameters and system parameters;
step S2, acquiring phonon deviation intensity based on the system parameters;
step S3, iteratively updating the phonon deviation intensity based on a linear Boltzmann equation and a phonon interface condition, obtaining a temperature corresponding to any phonon deviation intensity based on a ratio of any phonon deviation intensity to a basic definition of temperature, obtaining a pseudo temperature corresponding to any phonon deviation intensity based on a ratio of any phonon deviation intensity to a basic definition of pseudo temperature, and performing integral summation based on any phonon deviation intensity to obtain a heat flow corresponding to any phonon deviation intensity;
step S4, obtaining the difference value between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and defining the difference value as a relative error;
and step S5, comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
2. The method of resolving phonon thermal transport at an interface of claim 1, wherein: the method further comprises the following steps:
step S6, obtaining the calculation duration of the iterative update of the phonon deviation intensity, and defining the calculation duration as a first calculation duration;
step S7, if the first calculated time length is less than the preset total calculated time length, continuing to execute the steps S3, S4, S5, S6 and S7; otherwise, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
3. The method of resolving phonon thermal transport at an interface of claim 1 or 2, wherein:
the setting parameters comprise: space step length, system scale, time step length, total calculation time length and convergence error;
the system parameters include: temperature and pseudo-temperature;
the phonon bias intensities include: a first deviation intensity obtained based on the phonon intensity and a second deviation intensity obtained based on the phonon pseudo-equilibrium intensity.
4. The method of resolving phonon thermal transport at an interface of claim 3, wherein: the step S2 includes obtaining the first deviation intensity based on the phonon intensity;
the step of obtaining the first deviation strength specifically includes the following steps:
step S21, setting the initial value of the temperature as T0And acquiring the phonon intensity as follows based on the initial value of the temperature:
in the formula (I), the compound is shown in the specification,
it is the intensity of the phonon that is,
which is the velocity of the phonon group,
in order to approximate the planck constant,
in order to be the angular frequency of the phonons,pin order to be a phonon branch,
is the distribution of the phonons and the phonons,Dis the density of the phonon states,
representing a phonon branch aspAnd the phonon angular frequency is
Phonon state density of (a);
step S22, based on the phonon intensity, acquiring a phonon reference balance intensity as follows:
in the formula (I), the compound is shown in the specification,
for the phonon reference to the equilibrium intensity,
to take into account a constant type reference temperature
The bose-einstein distribution of (a);
step S23, obtaining a first deviation intensity based on the phonon intensity and the phonon reference balance intensity:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
it is the intensity of the phonon that is,
the equilibrium intensity is referred to as the phonon.
5. The method of resolving phonon thermal transport at an interface of claim 4, wherein: the step S2 further includes obtaining the second deviation intensity based on the phonon pseudo-balance intensity;
the step of obtaining the second deviation strength specifically includes the following steps:
step S24, setting the initial value of the pseudo temperature to be equal to the initial value of the temperature and to be T0And acquiring the phonon pseudo-equilibrium intensity based on the initial value of the pseudo-temperature as follows:
in the formula (I), the compound is shown in the specification,
for the pseudo-equilibrium intensity of the phonons,
is to take into account the initial pseudo-temperature
The bose-einstein distribution of (a);
step S25, obtaining a second deviation intensity based on the phonon intensity and the phonon pseudo-equilibrium intensity:
in the formula (I), the compound is shown in the specification,
is the second deviation intensity.
6. The method of resolving phonon thermal transport at an interface of claim 5, wherein:
the linear boltzmann equation is:
in the formula (I), the compound is shown in the specification,
in order to be the first deviation strength,
in order to be the second deviation strength,
in order to be the phonon relaxation time,
the angular frequency of the phonons is represented,pthe number of phonon branches is represented,
which represents the velocity of the phonon group,
representing a phonon branch aspAnd the phonon angular frequency is
(ii) a phonon group velocity of;
the interface conditions are as follows:
wherein A, B respectively represent two materials,
is the included angle between the normal vector of the interface and the group velocity, the positive direction of the normal direction of the interface points to the material B from the material A,
and
spectral interface transmission coefficients from material a to material B and from material B to material a respectively,
is the first off-set strength of material B,
is the first deflection strength of material a.
7. The method of resolving phonon thermal transport at an interface of claim 6, wherein: the obtaining of the corresponding temperature, pseudo-temperature, and heat flow based on any phonon bias intensity specifically includes:
step S31, obtaining a linear expression of temperature based on the basic definitions of the phonon deviation intensity and temperature:
in the formula (I), the compound is shown in the specification,
，
to take into account the temperatureTThe glass-einstein distribution at the time,
is a constant type of reference temperature, and,
in the form of a polar coordinate, the position of the lens,
in order to be the azimuth angle,
representing phonon branchespThe maximum phonon angular frequency of (a) is,
which is the velocity of the phonon group,
in order to be the first deviation strength,
the volume heat capacity of the frequency spectrum, namely the volume heat capacity of each phonon branch and a unit phonon angular frequency interval;
wherein the temperature is defined essentially as:
step S32, obtaining a linear expression of the pseudo temperature based on the phonon deviation intensity and the basic definition of the pseudo temperature:
in the formula:
in order to be the pseudo-temperature,
is the phonon relaxation time;
wherein the pseudo-temperature is substantially defined as:
step S33, acquiring a linear expression of the obtained heat flow based on the phonon deviation intensity:
in the formula (I), the compound is shown in the specification,qis the heat flow.
8. The method of resolving phonon thermal transport at an interface of claim 7, wherein: the step S5 specifically includes:
step S51, obtaining a relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:
in the formula (I), the compound is shown in the specification,
the relative error is indicated in the form of,nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,Tis the temperature;
and step S52, if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
9. The method of resolving phonon thermal transport at an interface of claim 7, wherein: the step S5 specifically includes:
step S53, obtaining a difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and defining the difference as a relative error:
in the formula (I), the compound is shown in the specification,
is a relative error;nrepresenting a spatial cell index;Nrepresents the total number of spatial units;ithe index of the number of iterations is represented,qis a heat flow;
and step S54, comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, the pseudo temperature and the heat flow corresponding to the current phonon deviation intensity.
10. A computer-readable storage medium characterized by: the computer readable storage medium stores a program which, when executed by a processor, causes the processor to perform the steps of the method according to any one of claims 1 to 9.
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| CN110275733A (en) * | 2019-06-27 | 2019-09-24 | 上海交通大学 | The GPU parallel acceleration method of phonon Boltzmann equation is solved based on finite volume method |
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