CN107844657B - Zinc oxide piezoresistor microscopic characteristic simulation optimization calculation model - Google Patents
Zinc oxide piezoresistor microscopic characteristic simulation optimization calculation model Download PDFInfo
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Abstract
A zinc oxide piezoresistor microscopic characteristic simulation optimization calculation model comprises a first optimization process and a second optimization process, and a microstructure simulation model based on a Voronoi grid, a conduction mechanism model of a crystal boundary double Schottky barrier and a crystal boundary partition model considering an intercrystalline phase bypass effect are established based on the first optimization process and the second optimization process. The beneficial effects are as follows: the most important characteristic of the new model is that the real conduction mechanism of the nonlinear grain boundary is reflected, and the core calculation parameters in the model are all inherent physical parameters of the ZnO varistor material, thereby truly realizing the calculable simulation of the correlation between the inherent microstructure and grain boundary characteristics of the ZnO varistor and the macroscopic electrical performance parameters.
Description
Technical Field
The invention relates to the research field of a novel high-performance ZnO varistor with a high voltage gradient, in particular to a micro-characteristic simulation optimization calculation model of a zinc oxide varistor.
Background
The most basic premise of material design is that the actual process of material experimental study and the various elements involved therein can be described by a suitable mathematical model; the most basic tool for material design is a calculation simulation model capable of truly reflecting the change rule of material characteristics.
Since the original optimization variables (raw material component formula and processing process conditions) and the final optimization target (macroscopic electrical performance parameters) in the optimization problem have a very complex correlation relationship, experimental research is almost the only tool and way for optimization solution for a long time. The important achievements in the aspect of optimizing the performance of the ZnO piezoresistor are basically established on the basis of a great amount of experimental research, and the important achievements are also the main reasons for the breakthrough of the novel high-performance ZnO valve plate in years of tracking research in China.
There is a very complex association between the optimization variables and the optimization objectives of the original optimization problem that is difficult to systematically generalize and describe. Compared with this, in each optimization problem divided into two steps, the original optimization variables and the intermediate optimization targets, and the association relationship between the intermediate optimization variables and the final optimization targets are definitely simpler and clearer. For the optimization variables and the optimization targets of each step in the optimization process divided into two steps, clear correlation relations between the optimization variables and the optimization targets can be established, and the correlation relations are basic conditions for further solving the original complete optimization problem.
For the calculation simulation model and algorithm of the ZnO varistor, some researchers have carried out related research and achieved certain results at present. However, the existing research results of calculating and simulating the ZnO piezoresistor still have certain problems and disadvantages, and it is difficult to obtain a direct correlation between the optimization variables and the target, that is, an intricate influence relationship and a correlation mechanism between the intrinsic microstructure and grain boundary characteristic parameters of the ZnO piezoresistor and macroscopic electrical performance parameters.
Disclosure of Invention
The invention aims to solve the problems and designs a zinc oxide piezoresistor microscopic characteristic simulation optimization calculation model. The specific design scheme is as follows:
a zinc oxide piezoresistor microscopic characteristic simulation optimization calculation model comprises a first optimization process and a second optimization process, and is characterized in that a microstructure simulation model based on a Voronoi grid, a conduction mechanism model of a grain boundary double Schottky barrier and a grain boundary partition model considering an intercrystalline phase bypass effect are established based on the first optimization process and the second optimization process,
in the first step of optimization process, the optimization variables are raw material component formula and processing process conditions, namely the original optimization variables of the original optimization problem; the optimization target of the first step optimization process is the microstructure and the grain boundary characteristic parameters of the ZnO piezoresistor and is also the optimization variable of the second step optimization process,
in the second optimization process, the optimization variables are microstructure and grain boundary characteristic parameters of the ZnO piezoresistor, namely the optimization target of the first optimization process; the optimization target of the second step optimization process is the macroscopic electrical performance parameter of the ZnO varistor, namely the final optimization target of the original optimization problem,
the establishing process of the microstructure simulation model based on the Voronoi grid comprises the steps of generating basic parameters of Voronoi seeds, confirming actual coordinates of the seeds and carrying out equivalent treatment on the actual coordinates, wherein the basic parameters, the actual coordinates and the equivalent treatment on the actual coordinates of the Voronoi seeds are sequentially carried out,
the model of the conduction mechanism of the grain boundary double Schottky barrier comprises basic formula establishment, model establishment and electron thermal excitation formula calculation, wherein the basic formula establishment, the model establishment and the electron thermal excitation formula calculation are carried out in the same way,
the grain boundary partition model considering the intergranular phase bypass effect comprises model structure type area determination and model formula determination, and the model structure type area determination and the model formula determination are carried out in sequence.
A Voronoi mesh is a vertically bisecting network graph in geometry. For n points on the euclidean plane (called Voronoi seeds), the Voronoi polygon corresponding to each seed point is defined as the set of all points on the plane that are less distant from the seed point than the other seed points; and the Voronoi polygons corresponding to all the seed points form corresponding Voronoi grids.
The specific shape of the Voronoi mesh is actually determined by the position distribution of the Voronoi seeds, and the Voronoi seeds distributed at different positions form corresponding Voronoi meshes with different shapes. The relevant algorithm of the previous researchers is necessarily improved in the basic parameters for generating the Voronoi seeds,
the basic parameters for generating Voronoi seeds include: average grain size S, in μm; average number of crystal grains N in horizontal X-axis directionXAverage number of crystal grains N in vertical Y-axis directionY(ii) a Degree of grain disorder D.
It is confirmed that in the actual coordinates of the seed,
will NX×NYThe Voronoi seeds are uniformly distributed according to the position of the central point of the honeycomb regular hexagon array,
for the ith column, i.e., in the positive X-direction, the coordinates of the Voronoi seed (X) are ordered for the jth row, i.e., in the positive Y-directionij 0,Yij 0) The calculation formula is as follows:
setting the maximum coordinate range (X) of the effective distribution area of the Voronoi gridmax,Ymax) And then:
Ymax＝S·(NY+1/2)
superposing random offset (dX) related to disorder degree D on the basis of uniformly distributed seed coordinatesij,dYij) And then:
dXij＝k·S·D·cos α
dYij＝k·S·D·sin α
k is random numbers which are uniformly distributed in the range of [0,1], α is random numbers which are uniformly distributed in the range of [0,2 pi ], the larger the value of the disorder degree D is, the larger the limit value of the random offset of the Voronoi seed is, and finally, the actual Voronoi seed coordinate is as follows:
and (3) the actual coordinates of the Voronoi seeds after the disorder degree related random offset is superposed are possibly out of the coordinate range of the effective distribution area of the Voronoi grid, and for the situation, new random numbers are regenerated to generate new random offset coordinates for calculation until all the actual coordinates of the Voronoi seeds are in the effective distribution area. Accordingly, any one Voronoi seed may appear at any position within the effective distribution area after superimposing a random offset amount as long as the product of the disorder D and the average size S of the grains reaches the length of the diagonal of the effective distribution area. If the product exceeds the length of the diagonal line of the effective distribution area, the actual distribution situation of the Voronoi seeds can not change substantially due to the limitation of the coordinate range of the effective distribution area, but the generated Voronoi seed coordinates can be caused to appear in the effective distribution areaThe probability of needing regeneration outside the distribution area increases. Therefore, the maximum value D for the degree of disorder in the actual algorithmmaxLimiting, the influence of the disorder degree D exceeding the upper limit value on the Voronoi seed distribution is actually equal to the upper limit value DmaxDmaxIs exactly the same but less computationally efficient, so with DmaxPerforming equivalent processing, wherein in the actual coordinate equivalent processing, a processing formula is as follows:
by adopting the algorithm, N in the effective distribution area can be obtainedX×NYAnd (3) coordinate data of each Voronoi seed, wherein each seed is superposed with random offset related to disorder degree D on the basis of the central point of the uniformly distributed honeycomb regular hexagon array. If N in the effective distribution area is directly adoptedX×NYAnd the Voronoi seeds are used for constructing the Voronoi grid, so that the Voronoi polygon generated by the seeds closest to the boundary of the effective distribution area has a part of area which is present outside the effective distribution area, and the part of the side line of the polygon has a divergent radial shape and generates a crossing phenomenon with the boundary.
Auxiliary Voronoi seeds are arranged outside the effective distribution area, the auxiliary Voronoi seeds are respectively distributed along four boundaries of the effective distribution area and form axial symmetry with the internal Voronoi seeds, the number of the auxiliary seeds on each side is NXOr NYTwice, it is thus possible to make the edges of the finally generated Voronoi polygon close to the boundaries fall exactly on the four boundaries of the effective distribution area without the phenomenon of intersection with the boundaries, which is essentially determined by the vertically bisecting property of the Voronoi polygon.
After the original Voronoi seeds inside the effective distribution area and the auxiliary Voronoi seeds outside the effective distribution area are jointly constructed to generate the Voronoi grids, the Voronoi polygons outside the effective distribution area are discarded, and only the Voronoi polygons inside the effective distribution area are reserved, namely the Voronoi polygons are used as a calculation simulation model of the ZnO piezoresistor microstructure.
In ZnO varistorsIn the microstructure, two adjacent ZnO crystal grains and a crystal boundary layer between the two ZnO crystal grains form a basic double Schottky barrier unit, and can be described by adopting a related theory of semiconductor physics. EVTop valence band energy level, E, of ZnO grainsCIs the conduction band bottom energy level of ZnO crystal grains, EFGFermi level (E) of ZnO crystal grainFBThe fermi level of the grain boundary layer. In the same thermal equilibrium system, electrons in all positions should have the same fermi level. In the initial unbalanced state, the Fermi level EFG of the n-type semiconductor ZnO crystal grains is higher than the Fermi level E of the grain boundary layerFBTherefore, free electrons in the ZnO crystal grains enter a crystal boundary layer through thermal movement until two sides reach equal Fermi level EF and enter an equilibrium state. The top energy level E of the valence band of ZnO crystal grainsVAs a relative energy level zero coordinate point.
In the grain boundary layer formed by the action of various additives, a large number of surface states and internal electron traps exist, free electrons from ZnO grains can be captured, a negative space charge layer is generated, energy bands of the ZnO grains adjacent to two sides of the grain boundary are bent upwards, and therefore two back-to-back grain boundary barriers, namely double Schottky barriers, are formed. Along with the migration of free electrons to the grain boundary, a corresponding electron depletion layer with a certain width is generated in the ZnO grain.
In the establishment of the basic formula (i),
the basic formula is established by using the Poisson equation:
wherein phiB(x) Is a function of the height distribution of the grain boundary barrier, and ρ (x) is a function of the charge distribution,. epsilon0And epsilon are the vacuum dielectric constant and the relative dielectric constant of ZnO respectively,
in the state without applied voltage (zero bias state), the ZnO crystal boundary barrier height phiB0And depletion layer width L0Donor density N mainly from ZnO grainsdAnd density of surface states N of grain boundariesiDetermining:
L0＝Ni/Nd
wherein e is the electronic charge. Donor density N for ZnO grainsdDonors including shallow and deep levels.
Because the number and the effect of the shallow level donors are far greater than those of the deep level donors (the difference in number is 1-2 orders of magnitude), an absolute dominant effect is achieved, the deep level donor density is not considered in the subsequent related calculation and derivation process of the paper, and only the shallow level donor density is considered.
Under the action of an external voltage, the double Schottky barrier is biased and generates continuous current, and ZnO grains on the left side of the barrier flow through grain boundaries and enter ZnO grains on the right side. With regard to the conduction mechanism of the double schottky barrier under the action of an applied voltage, in the establishment of a model,
under the action of an applied voltage, the following formula can be deduced from a Poisson equation:
wherein V is the voltage applied to both sides of the potential barrier, QiIs the charge density of the surface state filling. Filling charge density Q for surface statesi，
The derivation is based on the energy distribution function of the surface states:
wherein N isi(E) Is a function of the energy distribution of the surface state,
as a function of the impact distribution profile:
Ni(E)＝Ni·δ(E-Ei)
fermi distribution function fi(E) Comprises the following steps:
wherein k isBIs Boltzmann constant and T is absolute temperature ξiIs a quasi-Fermi level, which is a value obtained by shifting the Fermi level ξ under the action of an applied voltage V,
then it is the Fermi level ξ of the surface state in the neutral state, i.e., in the virtual initial statei nComprises the following steps:
the method is used for describing and solving the change characteristics of the ZnO varistor grain boundary double Schottky barrier such as barrier height phi B and surface state filling charge density Qi under the action of an external voltage V. A simulation calculation result of a certain practical example of a ZnO varistor grain boundary potential equation set is given: along with the increase of the external voltage V, the surface state filling charge density Qi is gradually increased, at the moment, the potential barrier height phi B is slowly reduced until all the surface states are completely filled, the potential barrier height phi B is rapidly reduced, and correspondingly, the ZnO piezoresistor grain boundary enters the breakdown conduction process.
In the calculation of the formula of the electron thermal excitation,
the current density J flowing through the double Schottky barrier is calculated by adopting an electron thermal excitation formula, and the comprehensive effect of hot electron movement in positive and negative directions is considered, wherein the calculation formula is as follows:
wherein A is Richardson constant, kBBoltzmann constant, T is absolute temperature, ∈ ξ is the energy level difference between the conduction band energy level Ec and the fermi energy level ξ, V is the voltage applied to the grain boundary double schottky barrier of the ZnO varistor,
J(V)＝(Vg-V)/(ρg·lg)
wherein, VgIs a single complete crystal boundary (including crystal) of the ZnO varistorThe boundary barrier portion and ZnO grains on both sides of the grain boundary), ρgAnd lgZnO grain resistivity and size, respectively.
The results obtained after considering the ZnO grain resistance are significantly different from those obtained without considering the ZnO grain resistance, particularly in the inversion region portion of the grain boundary barrier voltage characteristic curve.
Compared with the prior empirical formula which is adopted by researchers in a calculation simulation model and is subjected to simplification and approximation treatment, the series of equations is undoubtedly more complex, but the real conduction mechanism of the ZnO varistor grain boundary potential barrier is described more clearly and accurately.
The double Schottky barrier conduction mechanism model can well explain most experimental phenomena related to the ZnO piezoresistor, is not consistent with individual experimental results, and mainly is an experimental phenomenon that after an external voltage reaches a certain value, the grain boundary capacitance is subjected to abrupt jump increase, and then rapidly falls back. Therefore, related researchers successively put forward and develop a hole-induced breakdown mechanism to supplement and perfect the existing double Schottky barrier conduction mechanism model. The derivation calculation is carried out based on the hole-induced breakdown theory, and the action effect of the holes serving as minority carriers needs to be further introduced on the basis of the above introduced equation set describing the ZnO varistor grain boundary double Schottky barrier conduction mechanism model. After the hole effect is introduced, an equation set for describing a double Schottky barrier conduction mechanism model becomes more complex, and in a corresponding simulation calculation result, the capacitance characteristic parameter of the grain boundary barrier is greatly changed, so that a result basically consistent with measured data can be obtained, and the integral volt-ampere characteristic curve of the grain boundary barrier is not obviously influenced.
The model structure type region determination comprises a thick intergranular phase region, a double Schottky barrier region and a grain direct contact region,
the thick intergranular phase region: when the intergranular phase is thick, it is considered that a double schottky barrier cannot be formed, and a nonlinear volt-ampere characteristic is generated. For the conduction model of the region, only the impedance of the intergranular phase is considered, and the double Schottky barrier nonlinear impedance unit is not available any more. The proportion of the area to the total area of the single grain boundaries is defined as PA.
The double Schottky barrier region: the grain boundary part of the double schottky barrier is a layer of very thin and even no obvious observable intercrystalline phase, and the corresponding intercrystalline phase resistance has no obvious influence on the barrier conduction process and should not be considered any more. For the conduction model of the region, only the nonlinear impedance of the double schottky barrier is considered, and no intergranular phase impedance unit exists. The proportion of the area to the total area of the single grain boundaries is defined as PB.
The grain direct contact area: partition types not involved in the Eda model; defining the proportion of the area to the total area of single crystal boundary as PC。
ZnO crystal grains are divided according to different partition proportions of crystal boundary, and after each part of crystal grain resistance is connected with corresponding crystal boundary partition resistance in series, the parallel connection relation is integrally formed. The conductive characteristic of the entirety of the single grain boundary is all the above-described resistance element strings.
In the determination of the model formula, the determination formula is as follows:
wherein, the nonlinear impedance Z corresponding to the double Schottky barrier partDB＝VDB/IDB，
ZILImpedance corresponding to the thick intergranular phase region of type A in the grain boundary partition, ZGA1、ZGA2Impedance, Z, of ZnO crystal grain portions on the left and right sides corresponding to the regionDBIs the nonlinear impedance, Z, corresponding to the B-type double Schottky barrier region in the grain boundary partitionGB1、ZGB2Impedance, Z, of ZnO crystal grain portions on the left and right sides corresponding to the regionGC1、ZGC2Is the resistance of ZnO crystal grain parts at the left and right sides corresponding to the C type crystal grain direct contact area.
In addition to the resistive elements, there are also corresponding capacitive elements. For accurate description of the capacitance element, an expression related to hole-induced breakdown needs to be introduced into a model; on the other hand, the alternating current and impact characteristics of the ZnO piezoresistor closely related to the capacitive element have the characteristic of time-domain dynamic response, so that a corresponding equivalent circuit needs to be described by adopting a partial differential equation system, and a corresponding calculation simulation algorithm is also greatly different from a calculation simulation algorithm of direct current volt-ampere characteristics.
The zinc oxide piezoresistor microscopic characteristic simulation optimization calculation model obtained by the technical scheme has the beneficial effects that:
by applying a mathematical language of an optimization principle theory, the experimental research process of the ZnO piezoresistor is a typical multivariable and multi-objective optimization problem, and the experimental research is almost a unique tool and a unique way for optimization solution for a long time. By converting the original optimization problem into an equivalent optimization problem divided into two steps, a faster and more effective optimization solution idea and method can be obtained.
The most important characteristic of the new model is that the real conduction mechanism of the nonlinear grain boundary is reflected, and the core calculation parameters in the model are all inherent physical parameters of the ZnO varistor material, thereby truly realizing the calculable simulation of the correlation between the inherent microstructure and grain boundary characteristics of the ZnO varistor and the macroscopic electrical performance parameters.
Drawings
FIG. 1 is a mathematical description of the ZnO varistor research process of the present invention;
FIG. 2 is a schematic view of a Voronoi grid in accordance with the present invention;
FIG. 3 is a schematic view of the Voronoi seed coordinate calculation of the present invention;
FIG. 4 is a schematic diagram of the Voronoi grid construction using the auxiliary seeds according to the present invention;
FIG. 5 is a diagram of the initial state of the double Schottky barrier model of the ZnO varistor of the present invention;
FIG. 6 is a diagram of the equilibrium state of the double Schottky barrier model of the ZnO varistor of the present invention;
FIG. 7 is a bias diagram of the double Schottky barrier of the present invention under an applied voltage;
FIG. 8 is a graph of the change in barrier height for the dual Schottky barrier height and surface state fill charge density according to the present invention;
FIG. 9 is a graph of the change in fill charge for the dual Schottky barrier height and surface state fill charge density according to the present invention;
FIG. 10 is a voltage-current characteristic diagram of the ZnO varistor grain boundary barrier irrespective of grain resistance according to the present invention;
FIG. 11 is a voltage-current characteristic diagram of the ZnO varistor grain boundary barrier in consideration of grain resistance;
FIG. 12 is a diagram of a grain boundary partitioning model according to the present invention;
FIG. 13 is an equivalent circuit schematic diagram of the grain boundary partition model of the present invention;
FIG. 14 is a graph of the AC response of the ZnO varistor of the present invention;
FIG. 15 is a diagram of a thick intergranular phase region ratio PA according to the present invention;
fig. 16 is a diagram of the double schottky barrier region ratio PB according to the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings.
FIG. 1 is a mathematical description diagram of a research process of a ZnO varistor, as shown in FIG. 1, a zinc oxide varistor microscopic characteristic simulation optimization calculation model comprises a first optimization process and a second optimization process, and a microstructure simulation model based on a Voronoi grid, a conduction mechanism model of a grain boundary double Schottky barrier and a grain boundary partition model considering an intercrystalline phase bypass effect are established based on the first optimization process and the second optimization process,
in the first step of optimization process, the optimization variables are raw material component formula and processing process conditions, namely the original optimization variables of the original optimization problem; the optimization target of the first step optimization process is the microstructure and the grain boundary characteristic parameters of the ZnO piezoresistor and is also the optimization variable of the second step optimization process,
in the second optimization process, the optimization variables are microstructure and grain boundary characteristic parameters of the ZnO piezoresistor, namely the optimization target of the first optimization process; the optimization target of the second step optimization process is the macroscopic electrical performance parameter of the ZnO varistor, namely the final optimization target of the original optimization problem,
the establishing process of the microstructure simulation model based on the Voronoi grid comprises the steps of generating basic parameters of Voronoi seeds, confirming actual coordinates of the seeds and carrying out equivalent treatment on the actual coordinates, wherein the basic parameters, the actual coordinates and the equivalent treatment on the actual coordinates of the Voronoi seeds are sequentially carried out,
the model of the conduction mechanism of the grain boundary double Schottky barrier comprises basic formula establishment, model establishment and electron thermal excitation formula calculation, wherein the basic formula establishment, the model establishment and the electron thermal excitation formula calculation are carried out in the same way,
the grain boundary partition model considering the intergranular phase bypass effect comprises model structure type area determination and model formula determination, and the model structure type area determination and the model formula determination are carried out in sequence.
Fig. 2 is a schematic view of the Voronoi mesh of the present invention, and as shown in fig. 2, the Voronoi mesh is a vertically halved network graph in geometry. For n points on the euclidean plane (called Voronoi seeds), the Voronoi polygon corresponding to each seed point is defined as the set of all points on the plane that are less distant from the seed point than the other seed points; and the Voronoi polygons corresponding to all the seed points form corresponding Voronoi grids.
The specific shape of the Voronoi mesh is actually determined by the position distribution of the Voronoi seeds, and the Voronoi seeds distributed at different positions form corresponding Voronoi meshes with different shapes. The relevant algorithm of the previous researchers is necessarily improved in the basic parameters for generating the Voronoi seeds,
the basic parameters for generating Voronoi seeds include: average grain size S, in μm; average number of crystal grains N in horizontal X-axis directionXAverage number of crystal grains N in vertical Y-axis directionY(ii) a Degree of grain disorder D.
Fig. 3 is a schematic diagram of Voronoi seed coordinate calculation according to the present invention, as shown in fig. 3, in confirming actual coordinates of a seed,
will NX×NYThe Voronoi seeds are arranged according to the position of the center point of the honeycomb regular hexagon arrayThe uniform distribution is carried out, and the uniform distribution is carried out,
for the ith column, i.e., in the positive X-direction, the coordinates of the Voronoi seed (X) are ordered for the jth row, i.e., in the positive Y-directionij 0,Yij 0) The calculation formula is as follows:
setting the maximum coordinate range (X) of the effective distribution area of the Voronoi gridmax,Ymax) And then:
Ymax＝S·(NY+1/2)
superposing random offset (dX) related to disorder degree D on the basis of uniformly distributed seed coordinatesij,dYij) And then:
dXij＝k·S·D·cos α
dYij＝k·S·D·sin α
k is random numbers which are uniformly distributed in the range of [0,1], α is random numbers which are uniformly distributed in the range of [0,2 pi ], the larger the value of the disorder degree D is, the larger the limit value of the random offset of the Voronoi seed is, and finally, the actual Voronoi seed coordinate is as follows:
the actual coordinates of the Voronoi seed, superimposed with the random offsets of the disorder-related random offsets, may fall outside the coordinate range of the effective distribution area of the Voronoi grid, for which case a new one is regeneratedAnd (4) generating a new random offset coordinate to calculate until all the actual coordinates of the Voronoi seeds fall within the effective distribution area. Accordingly, any one Voronoi seed may appear at any position within the effective distribution area after superimposing a random offset amount as long as the product of the disorder D and the average size S of the grains reaches the length of the diagonal of the effective distribution area. If the product exceeds the length of the diagonal line of the effective distribution area, the actual distribution situation of the Voronoi seeds can not be changed essentially due to the limitation of the coordinate range of the effective distribution area, but the probability that the generated Voronoi seed coordinates appear outside the effective distribution area and need to be regenerated is increased. Therefore, the maximum value D for the degree of disorder in the actual algorithmmaxLimiting, the influence of the disorder degree D exceeding the upper limit value on the Voronoi seed distribution is actually equal to the upper limit value DmaxIs exactly the same but less computationally efficient, so with DmaxPerforming equivalent processing, wherein in the actual coordinate equivalent processing, a processing formula is as follows:
by adopting the algorithm, N in the effective distribution area can be obtainedX×NYAnd (3) coordinate data of each Voronoi seed, wherein each seed is superposed with random offset related to disorder degree D on the basis of the central point of the uniformly distributed honeycomb regular hexagon array. If N in the effective distribution area is directly adoptedX×NYAnd the Voronoi seeds are used for constructing the Voronoi grid, so that the Voronoi polygon generated by the seeds closest to the boundary of the effective distribution area has a part of area which is present outside the effective distribution area, and the part of the side line of the polygon has a divergent radial shape and generates a crossing phenomenon with the boundary.
Fig. 4 is a schematic diagram of the Voronoi mesh construction using the auxiliary seeds according to the present invention, as shown in fig. 4,
configuring auxiliary Voronoi seeds outside the effective distribution area, wherein the auxiliary Voronoi seeds respectively follow four boundaries of the effective distribution area and the insideThe Voronoi seeds form axial symmetry distribution, and the number of the auxiliary seeds on each side is NXOr NYTwice, it is thus possible to make the edges of the finally generated Voronoi polygon close to the boundaries fall exactly on the four boundaries of the effective distribution area without the phenomenon of intersection with the boundaries, which is essentially determined by the vertically bisecting property of the Voronoi polygon.
After the original Voronoi seeds inside the effective distribution area and the auxiliary Voronoi seeds outside the effective distribution area are jointly constructed to generate the Voronoi grids, the Voronoi polygons outside the effective distribution area are discarded, and only the Voronoi polygons inside the effective distribution area are reserved, namely the Voronoi polygons are used as a calculation simulation model of the ZnO piezoresistor microstructure.
FIG. 5 is a diagram of the initial state of the double Schottky barrier model of the ZnO varistor of the present invention; fig. 6 is a diagram of a double schottky barrier model equilibrium state of the ZnO varistor according to the present invention, and as shown in fig. 5 and 6, in the microstructure of the ZnO varistor, two adjacent ZnO grains and a grain boundary layer therebetween form a basic double schottky barrier unit, which can be described by using a theory related to semiconductor physics. EVTop valence band energy level, E, of ZnO grainsCIs the conduction band bottom energy level of ZnO crystal grains, EFGThe Fermi level (Fermi level) of ZnO crystal grains, and the EFB is the Fermi level of a grain boundary layer. In the same thermal equilibrium system, electrons in all positions should have the same fermi level. In the initial unbalanced state, due to the Fermi level E of the n-type semiconductor ZnO crystal grainsFGFermi level E higher than grain boundary layerFBTherefore, free electrons in ZnO crystal grains enter a crystal boundary layer through thermal movement until two sides reach equal Fermi level EFAnd entering an equilibrium state. In the subsequent calculation derivation process of the paper, the valence band top energy level EV of ZnO grains is taken as a relative energy level zero coordinate point.
In the grain boundary layer formed by the action of various additives, a large number of surface states and internal electron traps exist, free electrons from ZnO grains can be captured, a negative space charge layer is generated, energy bands of the ZnO grains adjacent to two sides of the grain boundary are bent upwards, and therefore two back-to-back grain boundary barriers, namely double Schottky barriers, are formed. Along with the migration of free electrons to the grain boundary, a corresponding electron depletion layer with a certain width is generated in the ZnO grain.
In the establishment of the basic formula (i),
the basic formula is established by using the Poisson equation:
wherein phiB(x) Is a function of the height distribution of the grain boundary barrier, and ρ (x) is a function of the charge distribution,. epsilon0And epsilon are the vacuum dielectric constant and the relative dielectric constant of ZnO respectively,
in the state without applied voltage (zero bias state), the ZnO crystal boundary barrier height phiB0And depletion layer width L0Donor density N mainly from ZnO grainsdAnd density of surface states N of grain boundariesiDetermining:
L0＝Ni/Nd
wherein e is the electronic charge. Donor density N for ZnO grainsdDonors including shallow and deep levels.
Because the number and the effect of the shallow level donors are far greater than those of the deep level donors (the difference in number is 1-2 orders of magnitude), an absolute dominant effect is achieved, the deep level donor density is not considered in the subsequent related calculation and derivation process of the paper, and only the shallow level donor density is considered.
Fig. 7 is a bias diagram of the double schottky barrier under the action of an applied voltage, as shown in fig. 7, under the action of the applied voltage, the double schottky barrier is biased and generates a continuous current, and ZnO grains on the left side of the barrier flow through grain boundaries and enter ZnO grains on the right side. With regard to the conduction mechanism of the double schottky barrier under the action of an applied voltage, in the establishment of a model,
under the action of an applied voltage, the following formula can be deduced from a Poisson equation:
wherein V is the voltage applied to both sides of the potential barrier, QiIs the charge density of the surface state filling. Filling charge density Q for surface statesi，
The derivation is based on the energy distribution function of the surface states:
wherein N isi(E) Is a function of the energy distribution of the surface state,
as a function of the impact distribution profile:
Ni(E)＝Ni·δ(E-Ei)
fermi distribution function fi(E) Comprises the following steps:
wherein k isBIs Boltzmann constant and T is absolute temperature ξiIs a quasi-Fermi level, which is a value obtained by shifting the Fermi level ξ under the action of an applied voltage V,
then it is the Fermi level ξ of the surface state in the neutral state, i.e., in the virtual initial statei nComprises the following steps:
FIG. 8 is a graph of the change in barrier height for the dual Schottky barrier height and surface state fill charge density according to the present invention; FIG. 9 is a graph showing the variation of the filling charge of the BiSchottky barrier height and the surface state filling charge density according to the present invention, as shown in FIGS. 8 and 9, for describing and solving the barrier height of the ZnO varistor grain boundary BiSchottky barrier under the action of the applied voltage VDegree phiBSurface state filling charge density QiThe change characteristic of (c). A simulation calculation result of a certain practical example of a ZnO varistor grain boundary potential equation set is given: surface state filling charge density Q with increasing applied voltage ViGradually increasing, at which point the barrier height phiBSlowly descending until the barrier height phi is reached after all surface states are completely filledBA rapid drop occurs and accordingly the ZnO varistor grain boundaries enter the breakdown conduction process.
In the calculation of the formula of the electron thermal excitation,
the current density J flowing through the double Schottky barrier is calculated by adopting an electron thermal excitation formula, and the comprehensive effect of hot electron movement in positive and negative directions is considered, wherein the calculation formula is as follows:
wherein A is Richardson constant, kBIs the Boltzmann constant, T is the absolute temperature, εξThe difference between the conduction band energy level Ec and the Fermi level ξ, V is the voltage applied to the ZnO varistor grain boundary double Schottky barrier,
J(V)＝(Vg-V)/(ρg·lg)
wherein, VgIs the voltage of single complete grain boundary (including the grain boundary barrier part and ZnO grains on two sides of the grain boundary) of the ZnO varistorgAnd lgZnO grain resistivity and size, respectively.
The results obtained after considering the ZnO grain resistance are significantly different from those obtained without considering the ZnO grain resistance, particularly in the inversion region portion of the grain boundary barrier voltage characteristic curve.
FIG. 10 is a voltage-current characteristic diagram of the ZnO varistor grain boundary barrier irrespective of grain resistance according to the present invention; fig. 11 is a voltage-current characteristic diagram of the ZnO varistor grain boundary barrier in consideration of grain resistance, and as shown in fig. 10 and 11, compared with the simplified and approximated empirical formula adopted by the prior researchers in the calculation simulation model, the series of equations is undoubtedly more complex, but the true conduction mechanism of the ZnO varistor grain boundary barrier is described more clearly and accurately.
The double Schottky barrier conduction mechanism model can well explain most experimental phenomena related to the ZnO piezoresistor, is not consistent with individual experimental results, and mainly is an experimental phenomenon that after an external voltage reaches a certain value, the grain boundary capacitance is subjected to abrupt jump increase, and then rapidly falls back. Therefore, related researchers successively put forward and develop a hole-induced breakdown mechanism to supplement and perfect the existing double Schottky barrier conduction mechanism model. The derivation calculation is carried out based on the hole-induced breakdown theory, and the action effect of the holes serving as minority carriers needs to be further introduced on the basis of the above introduced equation set describing the ZnO varistor grain boundary double Schottky barrier conduction mechanism model. After the hole effect is introduced, an equation set for describing a double Schottky barrier conduction mechanism model becomes more complex, and in a corresponding simulation calculation result, the capacitance characteristic parameter of the grain boundary barrier is greatly changed, so that a result basically consistent with measured data can be obtained, and the integral volt-ampere characteristic curve of the grain boundary barrier is not obviously influenced.
FIG. 12 is a view of a grain boundary partition model according to the present invention, as shown in FIG. 12, the model structure type region determination includes a thick intergranular phase region, a double Schottky barrier region, and a grain direct contact region,
the thick intergranular phase region: when the intergranular phase is thick, it is considered that a double schottky barrier cannot be formed, and a nonlinear volt-ampere characteristic is generated. For the conduction model of the region, only the impedance of the intergranular phase is considered, and the double Schottky barrier nonlinear impedance unit is not available any more. Defining the proportion of the area to the total area of single crystal boundary as PA。
The double Schottky barrier region: the grain boundary part of the double schottky barrier is a layer of very thin and even no obvious observable intercrystalline phase, and the corresponding intercrystalline phase resistance has no obvious influence on the barrier conduction process and should not be considered any more. For the conduction model of the region, only the nonlinear impedance of the double schottky barrier is considered, and no intergranular phase impedance unit exists. Defining the proportion of the area to the total area of single crystal boundary as PB。
The grain direct contact area: partition types not involved in the Eda model; defining the proportion of the area to the total area of single crystal boundary as PC。
FIG. 13 is an equivalent circuit schematic diagram of the grain boundary partition model of the present invention, as shown in FIG. 13, ZnO grains are divided according to the proportion of different partitions of the grain boundary, and after the resistance of each part of the grains is connected in series with the partition resistance of the corresponding grain boundary, the parallel connection relationship is formed integrally. The conductive characteristic of the entirety of the single grain boundary is all the above-described resistance element strings.
In the determination of the model formula, the determination formula is as follows:
wherein, the nonlinear impedance Z corresponding to the double Schottky barrier partDB＝VDB/IDB，
ZILImpedance corresponding to the thick intergranular phase region of type A in the grain boundary partition, ZGA1、ZGA2Impedance, Z, of ZnO crystal grain portions on the left and right sides corresponding to the regionDBIs the nonlinear impedance, Z, corresponding to the B-type double Schottky barrier region in the grain boundary partitionGB1、ZGB2Impedance, Z, of ZnO crystal grain portions on the left and right sides corresponding to the regionGC1、ZGC2Is the resistance of ZnO crystal grain parts at the left and right sides corresponding to the C type crystal grain direct contact area.
Fig. 14 is an ac response diagram of a ZnO varistor according to the present invention, and as shown in fig. 14, there are corresponding capacitive elements in addition to resistive elements. For accurate description of the capacitance element, an expression related to hole-induced breakdown needs to be introduced into a model; on the other hand, the alternating current and impact characteristics of the ZnO piezoresistor closely related to the capacitive element have the characteristic of time-domain dynamic response, so that a corresponding equivalent circuit needs to be described by adopting a partial differential equation system, and a corresponding calculation simulation algorithm is also greatly different from a calculation simulation algorithm of direct current volt-ampere characteristics.
Example 1
FIG. 15 shows the thick intergranular phase region ratio P according to the present inventionAA drawing;
FIG. 16 is a diagram showing the proportion PB of the double Schottky barrier region according to the present invention, and as shown in FIGS. 15 and 16, in a normal case, the grain boundary partition is mainly composed of a type A thick intergranular phase region and a type B double Schottky barrier region, and the respective proportions are PAAnd PB(ii) a The electrical characteristics of the whole grain boundary are the effects generated by the joint action of the A, B two types of regions, and have a remarkable correlation with the actual proportion of the two regions. When P is presentA<<PBIn the process, the B-type double Schottky barrier region has a dominant effect, and the whole grain boundary has obvious nonlinear characteristics, namely a 'good' grain boundary; when P is presentAAnd PBWhen the method is compared, the whole grain boundary still presents certain nonlinear characteristics, but the nonlinear coefficient is smaller, namely the 'poor' grain boundary; when P is presentA>>PBWhen the grain boundaries are substantially free of non-linear behavior throughout, i.e., "ohmic" grain boundaries.
Due to PA、PBMay be in the value of [0,1]]The variation within the interval is very large, and different samples are different in component formula and process condition, so that it can be expected that the parameter of each type of grain boundary in the grain boundary classification model has a very large variation range, for example, the data given in the related literature, the difference between the impedance of the ohmic grain boundary and the impedance of the good grain boundary is 2-5 orders of magnitude, the measured data of the total proportion of the ohmic grain boundary is 20% and 5-10% respectively, and the variation range of the total proportion of the good grain boundary is 15-60%, and the like. The grain boundary classification model performs approximation and simplification treatment on different grain boundary characteristics with high dispersity to a great extent, and the grain boundary classification model is classified into three grain boundary models with specific parameters for practical calculation simulation. In comparison, the grain boundary partition model can simulate the grain boundary characteristics more truly and reasonably.
Various characteristic parameters of ZnO varistor grain boundaries basically have the characteristic of normal distribution although the ZnO varistor grain boundaries have large dispersity. Therefore, the proportion of each partition in the grain boundary partition model proposed by the paper is also calculated by using the normal distribution functionAnd (6) simulating. For the thick intergranular phase region of A type in the grain boundary partition, defining a corresponding proportion parameter PAThe normal distribution model of (a) is,
the arithmetic mean value of the parameter distribution is the standard deviation of the parameter distribution. For each grain boundary in the actual calculation simulation model, generating P by using a standard normal distribution random number constructorANumerical values.
For the direct contact area of C type crystal grains in the grain boundary subarea, the corresponding proportion parameter PCBy reaction of a compound with PAThe definition and assignment are performed in a completely similar manner. In general, the proportion of the C-type region in the grain boundary is very small, so the set value is correspondingly small, and the values in the subsequent calculation examples are 0 and [10-9, 10-6%]. For a B-type double Schottky barrier region in the grain boundary partition, the corresponding proportion parameter PBThen can be represented by PA、PCAnd (4) carrying out calculation:
p obtained by the above formulaBNumerical values, also in accordance with the form of a normal distribution, with PAThe normal distribution curve is substantially axisymmetrically distributed. In the actual calculation simulation model, if P is randomly generatedA、PB、PCData sum exceeding [0,1]]And if so, discarding the invalid data and regenerating a new random number.
Actually, because the component formulas and the process conditions of different samples are different, the actual distribution conditions of each type of region in the grain boundary partition model can be greatly different, and the randomly generated distribution proportion data P of each type of region can be adjusted by changing related parameters involved in the modelA、PB、PCAverage and standard deviation of. For the data given in the above example, if P is setAIn [0,0.3 ]]In the range of "good" grain boundaries, (0.3, 0.7)]In the range of "poor" grain boundaries, (0.7,1]The grain boundaries of 'ohm' are in the range, the proportion of the grain boundaries of 'good', 'poor' and 'ohm' to the total number is 0.51, 0.37 and 0.12 respectively, and the actual measurement data published by part of researchers can be well matched; the above model may well describe other researchers to as well as changing the relevant calculation parametersAnd (4) outputting the measured data.
Example 2
In the microstructure of the ZnO varistor, in addition to the ZnO crystal grains and the grain boundaries thereof constituting the main crystal phase, there are some special forms of structures including pores, spinels, and the like, which, although much smaller in number and importance than the main crystal phase, have a certain influence on the macroscopic electrical properties of the ZnO varistor. On the basis of the ZnO piezoresistor calculation simulation model taking the Voronoi grid and grain boundary partition model as the core, the influence of various special structures can be simulated by properly improving and introducing some new parameters.
For the air holes, a certain proportion P can be randomly selectedPoreThe conductivity of the ZnO grains simulated by the Voronoi polygon of (1) is set to 0, that is, it can be equivalent to pores having insulating properties.
For the function of spinel, a spinel partition with insulating property can be introduced into the grain boundary partition model, and the proportion parameter P of the spinel partition in the total area of the grain boundary is setParticleThis parameter can be set to a fixed value, or a method similar to that used for other partitions can be used to generate random normal distribution data.
The technical solutions described above only represent the preferred technical solutions of the present invention, and some possible modifications to some parts of the technical solutions by those skilled in the art all represent the principles of the present invention, and fall within the protection scope of the present invention.
Claims (9)
1. A method for simulating, optimizing and calculating a model of microscopic characteristics of a zinc oxide piezoresistor comprises a first optimization process and a second optimization process, and is characterized in that a microstructure simulation model based on a Voronoi grid, a conduction mechanism model of a grain boundary double Schottky barrier and a grain boundary partition model considering an intercrystalline phase bypass effect are established based on the first optimization process and the second optimization process,
in the first step of optimization process, the optimization variables are raw material component formula and processing process conditions, namely the original optimization variables of the original optimization problem; the optimization target of the first step optimization process is the microstructure and the grain boundary characteristic parameters of the ZnO piezoresistor and is also the optimization variable of the second step optimization process,
in the second optimization process, the optimization variables are microstructure and grain boundary characteristic parameters of the ZnO piezoresistor, namely the optimization target of the first optimization process; the optimization target of the second step optimization process is the macroscopic electrical performance parameter of the ZnO varistor, namely the final optimization target of the original optimization problem,
the establishing process of the microstructure simulation model based on the Voronoi grid comprises the steps of generating basic parameters of Voronoi seeds, confirming actual coordinates of the seeds and carrying out equivalent treatment on the actual coordinates, wherein the basic parameters, the actual coordinates and the equivalent treatment on the actual coordinates of the Voronoi seeds are sequentially carried out,
the model of the conduction mechanism of the grain boundary double Schottky barrier comprises basic formula establishment, model establishment and electron thermal excitation formula calculation, wherein the basic formula establishment, the model establishment and the electron thermal excitation formula calculation are carried out in the same way,
the grain boundary partition model considering the intergranular phase bypass effect comprises model structure type area determination and model formula determination, and the model structure type area determination and the model formula determination are carried out in sequence.
2. The method for simulating and optimizing the computational model of the microscopic characteristics of the zinc oxide piezoresistor according to claim 1, wherein in the basic parameters for generating the Voronoi seeds,
the basic parameters for generating Voronoi seeds include: average grain size S, in μm; average number of crystal grains N in horizontal X-axis directionXAverage number of crystal grains N in vertical Y-axis directionY(ii) a Degree of grain disorder D.
3. The method for simulating and optimizing the computational model of the microscopic characteristics of the zinc oxide piezoresistor according to claim 1, wherein the actual coordinates of the seeds are determined,
will NX×NYThe Voronoi seeds are uniformly distributed according to the position of the center point of the honeycomb regular hexagon arrayThe mixture is evenly distributed and then is evenly distributed,
for the ith column, i.e., in the positive X-direction, the coordinates of the Voronoi seed (X) are ordered for the jth row, i.e., in the positive Y-directionij 0,Yij 0) The calculation formula is as follows:
setting the maximum coordinate range (X) of the effective distribution area of the Voronoi gridmax,Ymax) And then:
Ymax＝S·(NY+1/2)
superposing random offset (dX) related to disorder degree D on the basis of uniformly distributed seed coordinatesij,dYij) And then:
dXij＝k·S·D·cosα
dYij＝k·S·D·sinα
k is random numbers which are uniformly distributed in the range of [0,1], α is random numbers which are uniformly distributed in the range of [0,2 pi ], the larger the value of the disorder degree D is, the larger the limit value of the random offset of the Voronoi seeds is, and finally, the actual Voronoi seed coordinates are as follows:
4. the method for simulating and optimizing the computational model of the microscopic characteristics of the zinc oxide piezoresistor according to claim 1, wherein in the actual coordinate equivalent processing, the processing formula is as follows:
obtaining N in effective distribution areaX×NYAnd (3) coordinate data of each Voronoi seed, wherein each seed is superposed with random offset related to disorder degree D on the basis of the central point of the uniformly distributed honeycomb regular hexagon array.
5. The method for simulating and optimizing the computational model of the microscopic characteristics of the zinc oxide piezoresistor according to claim 1, wherein in the establishment of the basic formula,
the basic formula is established by using the Poisson equation:
wherein phiB(x) Is a function of the height distribution of the grain boundary barrier, and ρ (x) is a function of the charge distribution,. epsilon0And epsilon are the vacuum dielectric constant and the relative dielectric constant of ZnO respectively,
under the condition of no external voltage action, ZnO crystal boundary barrier height phiB0And depletion layer width L0Donor density N mainly from ZnO grainsdAnd density of surface states N of grain boundariesiDetermining:
L0＝Ni/Nd
wherein e is an electron charge, donor density N for ZnO crystal grainsdDonors including shallow and deep levels.
6. The method for simulating and optimizing the computational model of the microscopic characteristics of the zinc oxide piezoresistor according to claim 1, wherein in the model establishment,
under the action of an applied voltage, the following formula can be deduced from a Poisson equation:
wherein V is the voltage applied to both sides of the potential barrier, QiIs the charge density of the surface state filling, for which the charge density Qi，
The derivation is based on the energy distribution function of the surface states:
wherein N isi(E) Is a function of the energy distribution of the surface state,
as a function of the impact distribution profile:
Ni(E)＝Ni·δ(E-Ei)
fermi distribution function fi(E) Comprises the following steps:
wherein k isBIs Boltzmann constant and T is absolute temperature ξiIs a quasi-Fermi level, which is a value obtained by shifting the Fermi level ξ under the action of an applied voltage V,
then it is the Fermi level ξ of the surface state in the neutral state, i.e., in the virtual initial statei nComprises the following steps:
7. the method for simulating and optimizing the computational model of the microscopic characteristics of the zinc oxide piezoresistor according to claim 1, wherein in the calculation of the electron thermal excitation formula,
the current density J flowing through the double Schottky barrier is calculated by adopting an electron thermal excitation formula, and the comprehensive effect of hot electron movement in positive and negative directions is considered, wherein the calculation formula is as follows:
wherein A is Richardson constant, kBIs the Boltzmann constant, T is the absolute temperature, εξThe difference between the conduction band energy level Ec and the Fermi level ξ, V is the voltage applied to the ZnO varistor grain boundary double Schottky barrier,
J(V)＝(Vg-V)/(ρg·lg)
wherein, VgIs the voltage, rho, of a single complete grain boundary of a ZnO varistorgAnd lgZnO grain resistivity and size, respectively.
8. The method for simulating and optimizing the computational model of the microscopic characteristics of the zinc oxide piezoresistor according to claim 1, wherein the model structure type region determination comprises a thick intergranular phase region, a double Schottky barrier region and a grain direct contact region,
the thick intergranular phase region: when the intergranular phase is very thick, the double Schottky barrier cannot be formed, nonlinear volt-ampere characteristics are generated, for the conduction model of the region, only the impedance of the intergranular phase is considered, a double Schottky barrier nonlinear impedance unit is not needed any more, the proportion of the region to the total area of a single crystal boundary is defined as PA,
the double Schottky barrier region: the grain boundary part of the double Schottky barrier is a layer of very thin and even no obviously observable intercrystalline phase, the corresponding intercrystalline phase impedance has no obvious influence on the barrier conduction process, and should not be considered any more, for the conduction model of the region, only the nonlinear impedance of the double Schottky barrier is considered, no intercrystalline phase impedance unit exists, the proportion of the region in the total area of single grain boundary is defined as PB,
the grain direct contact area: partition types not involved in the Eda model; the ratio of the area to the total area of the single grain boundaries is defined as PC.
9. The method for simulating and optimizing the calculation model of the microscopic characteristics of the zinc oxide piezoresistor according to claim 1, wherein in the determination of the model formula, the determination formula is as follows:
wherein, the nonlinear impedance Z corresponding to the double Schottky barrier partDB＝VDB/IDB，
ZILImpedance corresponding to the thick intergranular phase region of type A in the grain boundary partition, ZGA1、ZGA2Impedance, Z, of ZnO crystal grain portions on the left and right sides corresponding to the regionDBIs the nonlinear impedance, Z, corresponding to the B-type double Schottky barrier region in the grain boundary partitionGB1、ZGB2Impedance, Z, of ZnO crystal grain portions on the left and right sides corresponding to the regionGC1、ZGC2Is the resistance of ZnO crystal grain parts at the left and right sides corresponding to the C type crystal grain direct contact area.
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