Method and system for predicting mobility of semiconductor carriers

Abstract

The invention relates to the technical field of semiconductors, in particular to a semiconductor carrier mobility prediction method and system, and the method comprises the steps: building a Markos jump rate which represents the transmission of charges in an organic semiconductor through a discontinuous jump mechanism; a main equation under the steady-state condition is established based on the Markos jump rate, and the main equation is used for representing that in the simulated charge transfer system, the charge occupation probability of each site depends on each other through the charge jump process and reaches balance under the steady state; establishing a hole mobility model and a mobility correction factor irrelevant to density, and establishing a mobility model based on the hole mobility model and the mobility correction factor; establishing a weak density dependency function based on the effective temperature, establishing an improved mobility model based on the weak density dependency function and the mobility model, and predicting the mobility of the semiconductor carrier based on the improved mobility model; the migration rate prediction error can be reduced, and the prediction precision can be improved.

Description

Technical Field

[0001] The present invention relates to the field of semiconductor technology, and in particular to a method and system for predicting the mobility of semiconductor carriers. Background Art

[0002] An accurate carrier mobility model for amorphous organic semiconductors is crucial for describing the characteristics of electronic devices. Although the extended Gaussian disorder model (EGDM) in related technologies effectively describes the dependence of mobility on temperature and carrier concentration and simplifies the lattice structure, certain limiting conditions do not conform to the established assumptions, resulting in errors. Summary of the Invention

[0003] The purpose of the present invention is to provide a method and system for predicting the mobility of semiconductor carriers, aiming to reduce errors and improve prediction accuracy by improving the weak density-dependent function and the mobility model.

[0004] To achieve the above purpose, the present invention provides the following technical solutions:

[0005] In a first aspect, an embodiment of the present invention provides a method for predicting the mobility of semiconductor carriers, the method comprising the following steps:

[0006] Establish a Marcus hopping rate that characterizes the transport of charges through a discontinuous hopping mechanism in an organic semiconductor;

[0007] Based on the Marcus hopping rate, establish a master equation under steady-state conditions, which is used to characterize that the charge occupation probabilities at each site in the simulated charge transport system are interdependent through the charge hopping process and reach equilibrium under steady state;

[0008] Establish a hole mobility model and a density-independent mobility correction factor, and establish a mobility model based on the hole mobility model and the mobility correction factor; wherein, the mobility correction factor is used to characterize the electric field dependence, and the hole mobility model is used to characterize the relationship between the hole mobility and the carrier concentration at different temperatures under zero electric field conditions;

[0009] Based on the effective temperature, establish a weak density-dependent function, establish an improved mobility model based on the weak density-dependent function and the mobility model, and predict the mobility of semiconductor carriers based on the improved mobility model; wherein, the effective temperature is used to characterize the temperature experienced by the carriers under the action of an electric field.

[0010] Optionally, the formula for the Marcus hopping rate is as follows:

[0011]

[0012] where W ijrepresents the hopping rate, represents the reduced Planck constant, k B represents the Boltzmann constant, T represents temperature, E r is a parameter representing the molecular reorganization energy, ΔE ij = E i - E j defines the energy difference between sites i and j, J ij represents the charge transfer integral between these two sites.

[0013] Optionally, the master equation under the steady-state condition is expressed as:

[0014]

[0015] where, p i represents the probability that site i is occupied by a charge, p j represents the probability that site j is occupied by a charge, W ij is the hopping rate of the charge hopping from site i to site j, W ji is the hopping rate of the charge hopping from site j to site i.

[0016] Optionally, the calculation formula of the hole mobility model is:

[0017] μ(T,p,F)≈μ(T,p)f(T,F) (3);

[0018]

[0019] where, μ(T,p,F) represents the hole mobility before improvement, μ(T,p) represents the intrinsic mobility that only depends on temperature and carrier concentration, f(T,F) is the mobility correction factor dependent on the electric field, μ0(T) represents the dimensionless scaling factor of the intrinsic mobility, which is used to adjust the mobility baseline value at zero carrier concentration (p→0), μ * represents the scaled intrinsic mobility, c = 0.42, which represents the exponential decay coefficient of the temperature dependence of the mobility; represents the temperature-normalized energy disorder, p represents the carrier concentration, b represents the lattice constant, δ represents the exponential adjustment factor of the influence of the carrier density on the mobility.

[0020] Optionally, the calculation formula of the mobility correction factor is:

[0021]

[0022] where, f(T,F) represents the mobility correction factor at the electric field F and temperature T, represents the temperature-normalized energy disorder, σ represents the energy disorder degree, ebF represents the energy modulation effect of the electric field on the carrier migration, and A and B represent the empirical fitting parameter factors.

[0023] Optionally, the weak density-dependent function is:

[0024]

[0025]

[0026] where T eff represents the effective temperature, γ is a numerical coefficient, k B represents the Boltzmann constant, e represents the elementary charge quantity, α represents the localization length, F represents the electric field, T represents the temperature; g(T eff , F) represents the weak density-dependent electric field modulation function, μ(T eff , p, F) represents the mobility expression improved by strengthening the electric field dependence through the exponential correction term, μ(T eff , p) represents the intrinsic mobility that only depends on the effective temperature and the carrier concentration, and c1 and c2 are weak density-dependent parameters.

[0027] In a second aspect, an embodiment of the present invention provides a mobility prediction system for semiconductor carriers, and the system includes:

[0028] At least one processor;

[0029] At least one memory for storing at least one program;

[0030] When the at least one program is executed by the at least one processor, the at least one processor implements the method described in any one of the above.

[0031] The beneficial effects of the present invention are as follows: First, the Marcus hopping rate, which characterizes the charge transport through a discontinuous hopping mechanism in an organic semiconductor, is established. Then, based on the Marcus hopping rate, a master equation under steady-state conditions is established. The master equation is used to characterize that in a simulated charge transport system, the charge occupancy probabilities at each site are interdependent through the charge hopping process and reach equilibrium under steady state. Then, a hole mobility model and a density-independent mobility correction factor are established, and a mobility model is established based on the hole mobility model and the mobility correction factor. Among them, the mobility correction factor is used to characterize the electric field dependence, and the hole mobility model is used to characterize the relationship between the hole mobility and the carrier concentration at different temperatures under zero electric field conditions. Finally, a weak density-dependent function is established based on the effective temperature, and an improved mobility model is established based on the weak density-dependent function and the mobility model. The mobility of semiconductor carriers is predicted based on the improved mobility model. The effective temperature is used to characterize the temperature experienced by the carriers under the action of an electric field. The improved mobility model of the present invention uses a calibrated effective temperature to replace the actual temperature, enhancing the dependence on the electric field, and thus more accurately reflecting the hopping transport law of disordered organic semiconductors through random spatial positions. By improving the weak density-dependent function and the mobility model, the present invention can reduce errors and improve the prediction accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required in the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0033] Figure 1 It is a schematic flowchart of the method for predicting the mobility of semiconductor carriers in an embodiment of the present invention;

[0034] Figure 2 It is a schematic diagram of the relationship between the hole mobility and the carrier concentration at different temperatures of the TCTA material under zero electric field conditions;

[0035] FIG. 3(a) is a schematic diagram of curves showing the influence of different temperatures on the mobility at low carrier concentration and high carrier concentration as the electric field changes in TCTA;

[0036] FIG. 3(b) is a schematic diagram of the corrected curve of FIG. 3(a) based on the modified equations;

[0037] Figure 4 It is a schematic diagram of the comparison between the experimental data and the theoretical prediction of the space charge limited current of the TCTA material changing with voltage;

[0038] Figure 5 Schematic diagram of current density-voltage characteristics of different thicknesses of TCTA at room temperature;

[0039] Figure 6 Schematic diagram of voltage varying with charge density at the position of x = 0 for a 175 nm thick TCTA material at 213 K and 295 K;

[0040] Figure 7 Schematic diagram of voltage varying with charge density for 303 nm and 102 nm thick TCTA layers at room temperature;

[0041] Figure 8 Schematic diagram of the distribution of carrier density along the position for devices with different thicknesses at low and high current densities in a hole-only device based on TCTA material at room temperature;

[0042] Figure 9 Schematic diagram of the calculated distribution of the electric field as a function of position x in a hole-type device based on TCTA at room temperature;

[0043] Figure 10 Schematic diagram of the structure of the mobility prediction system for semiconductor carriers in an embodiment of the present invention. Detailed implementation manners

[0044] The concept, specific structure and technical effects of the present invention will be clearly and completely described below in conjunction with the embodiments and the drawings to fully understand the purpose, scheme and effects of the present invention. It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0045] Refer to Figure 1 , the present invention provides a method for predicting the mobility of semiconductor carriers, and the method includes the following steps:

[0046] S100, establish the Marcus hopping rate that characterizes the transport of charges in an organic semiconductor through a non - continuous hopping mechanism;

[0047] In some embodiments, the formula of the Marcus hopping rate is as follows:

[0048]

[0049] Wherein, W ij represents the hopping rate, represents the reduced Planck constant, k B represents the Boltzmann constant, T represents the temperature, E r is a parameter representing the molecular reorganization energy, ΔE ij = E i - Ej Defines the energy difference, J, between sites i and j ij Represents the charge transfer integral between these two sites.

[0050] In some embodiments, the master equation under the steady-state condition is expressed as:

[0051]

[0052] Where p i Represents the probability that site i is occupied by a charge, and p j Represents the probability that site j is occupied by a charge, and W ij Is the hopping rate of the charge from site i to site j, and W ji Is the hopping rate of the charge from site j to site i.

[0053] It should be noted that charge transport in organic semiconductors occurs through a non-continuous hopping mechanism, which involves hopping between two weakly coupled molecular sites i and j. The hopping frequency is denoted by W ij And can be described by Marcus theory, which takes into account the energy configuration of the system and the hopping rate based on the local state energy. The formula for the Marcus hopping rate is as follows:

[0054]

[0055] Where W ij Represents the hopping rate, Represents the reduced Planck constant, k B Represents the Boltzmann constant, T represents the temperature, E r Is a parameter representing the molecular reorganization energy, ΔE ij = E i - E j Defines the energy difference, J, between sites i and j ij Represents the charge transfer integral between these two sites.

[0056] The energy difference ΔE ij Also includes a part affected by the electric field F applied in the x direction, i.e., -eFR ij,x Where R ij,x Represents the distance between sites along the x direction, and e represents the magnitude of the elementary charge. Thus, the electric field F affects the hopping rate W ij By influencing the energy difference ΔE ij , which ultimately affects the carrier mobility μ. The electric field changes the energy difference between sites, thereby changing the probability of electrons hopping between different sites.

[0057] S200, establish the master equation under steady-state conditions based on the Marcus hopping rate, which is used to characterize that in a simulated charge transport system, the charge occupation probabilities of each site are interdependent through the charge hopping process and reach equilibrium under steady state;

[0058] The mobility function μ is obtained by solving the steady-state master equation that describes the site occupation probability of charges in the simulation box. The master equation takes into account the strong Coulomb repulsion effect at the sites, so it does not allow two charges to occupy the same site simultaneously. We randomly select the site energies from a Gaussian distribution with a standard deviation of σ. The methods described in reference [1] in the related technology are used to randomly generate the site positions and the direct transfer integrals. The master equation (ME) describes how the charge occupation probabilities of each site in a simulated charge transport system are interdependent through the charge hopping process and reach equilibrium under steady state. By solving this equation, the mobility function μ of the system can be obtained, and then the movement of charge carriers inside the material under external conditions such as an electric field can be described.

[0059] Under steady-state conditions, the master equation (ME) can be expressed as:

[0060]

[0061] where, p i represents the probability that site i is occupied by a charge, p j represents the probability that site j is occupied by a charge, W ij is the hopping rate of the charge from site i to site j, and W ji is the hopping rate of the charge from site j to site i.

[0062] The factor (1 - p i ) takes into account the Coulomb repulsion to ensure that each site can be occupied by at most one charge; W ij p i (1 - p j ) and W ji p j (1 - p i ) calculate the expected number of hops from site i to j and from site j to i respectively; the equilibrium condition [W ij p i (1 - p j ) - W ji p j (1 - p i )] = 0 indicates that the net hopping rate between all site pairs is zero, meaning that the system has reached dynamic equilibrium.

[0063] S300, establish a hole mobility model and a density-independent mobility correction factor, and establish a mobility model based on the hole mobility model and the mobility correction factor; wherein, the mobility correction factor is used to characterize the electric field dependence, and the hole mobility model is used to characterize the relationship between the hole mobility and the carrier concentration at different temperatures under zero electric field conditions;

[0064] Figure 2 shows the relationship between the hole mobility μ and the carrier concentration p at different temperatures T under the condition of zero electric field (F = 0) for the TCTA material. The data points are obtained by calculating the randomly generated material structures and electron transfer probabilities in a simulated cubic box, and these results are averaged based on five different disordered material layouts. The size of the error bars is usually comparable to or smaller than the size of the data points. The curve is fitted by using the parametric equation (5a), where μ * = 6.1×10 -6 m 2 / Vs, c = 0.41.

[0065] In Figure 2 the influence of different temperatures T and carrier concentrations p on the result μ is studied. The symbols represent the master equation results of μ as a function of p at different values under the condition of electric field F = 0, for the TCTA material. The curve represents the result fitted by using the parametric scheme given in reference [1].

[0066] In some embodiments, the calculation formula of the hole mobility model is:

[0067] μ(T,p,F)≈μ(T,p)f(T,F)(3);

[0068]

[0069] wherein, μ(T,p,F) represents the hole mobility before improvement, μ(T,p) represents the intrinsic mobility that only depends on temperature and carrier concentration, f(T,F) is the mobility correction factor at electric field F and temperature T, μ0(T) represents the dimensionless scaling factor of the intrinsic mobility, which is used to adjust the mobility baseline value at zero carrier concentration (p→0), μ * represents the scaled intrinsic mobility, c = 0.42, which represents the exponential decay coefficient of the mobility temperature dependence; represents the temperature-normalized energy disorder, p represents the carrier concentration, b represents the lattice constant, and δ represents the exponential adjustment factor of the influence of carrier density on mobility.

[0070] Given μ * = 6.1×10 -6m 2 / Vs, c = 0.41. We note that the corrected value of c is very close to the value c = 0.42 used in Ref. [1] in previous simulations. Therefore, the dependence of T is very similar to that of EGDM. However, although the mobility models in Ref. [1] can well describe the charge transport at low densities, their fits in the high-density region are not satisfactory. This is because these mobility models only consider the non-Arrhenius temperature dependence ln(μ) ∝ 1 / T 2 (see Eq. (4)), and this dependence may no longer be applicable at high densities. Therefore, for the high-density region, the mobility model needs to be further corrected or additional mechanisms need to be introduced to more accurately describe the charge transport behavior.

[0071] Figure 3 depicts how the mobility μ at low and high carrier concentrations p is affected by different temperatures T as the electric field F varies in TCTA. The points in the figure represent the results of the master equation. The curves in Fig. 3(a) adopt Eqs. (3)-(6) with parameters A = 0.35 and B = 1.4 in Ref. [1]. While the curves in Fig. 3(b) are based on the modified set of equations.

[0072] In Fig. 3, the effect of the dimensionless electric field F on μ at low (main figure) and high (inset) p at different temperatures is shown. Among them on the left (a), the parameters are A = 0.35, B = 1.4. Notably, just like in EGDM in Ref. [1], a density-independent mobility correction factor f(T,F) is used to account for the electric field dependence. The compact parameterization deviates from the ME results in a large F range and is accurate enough within the experimentally relevant electric field range until Feb / σ ≈ 1. Obviously, when Feb > σ and σ > 1.7 / k B T, Eq. (5) cannot hold, the exponent becomes positive, and the equation predicts that the mobility grows exponentially with the increase of the electric field and gives a wrong infinite limit at any electric field. In the low electric field region, when lnμ is proportional to ABF 2 the AB factor characterizes the electric field sensitivity. For EGDM, the values A = 0.44 and B = 0.8 are used in the parameterization

[11] . This factor is equal to 0.49, while for EGDM it is 0.35. Therefore, it is concluded that the dependence of the mobility of TCTA on F is stronger.

[0073] In some embodiments, the calculation formula of the mobility correction factor is:

[0074]

[0075] where f(T,F) represents the mobility correction factor at the electric field F and temperature T, Represents the temperature-normalized energy disorder σ represents the energy disorder, ebF represents the energy modulation effect of the electric field on carrier migration, and A, B represent empirical fitting parameter factors.

[0076] In organic semiconductors, charge transport usually occurs through a hopping mechanism, where the transition of charges between different local states plays a key role. This hopping process is not only affected by the electric field but also related to the transport distance of charges in the material. Therefore, the spatial scale parameter α is particularly important here because it can reflect the degree of influence of the electric field on the hopping mobility. Mobility, as a measure of the average drift velocity of carriers under the action of an electric field, its variation is closely related to the spatial scale related to the electric field.

[0077] S400, establish a weak density-dependent function based on the effective temperature, and establish an improved mobility model based on the weak density-dependent function and the mobility model; wherein, the effective temperature is used to characterize the temperature experienced by carriers under the action of an electric field;

[0078] To better describe the dependence of carrier mobility on the electric field F, the effective temperature T eff (T,F) concept is introduced. The effective temperature is a theoretical tool used to describe the temperature experienced by carriers under the action of an electric field, which may be different from the physical temperature of the material. This parameter takes into account the influence of the electric field and the spatial scale parameter α on the energy state of carriers, thus more accurately reflecting the motion state of carriers in the electric field.

[0079] Although the localization length α is a key parameter affecting the hopping mobility, it does not appear in equation (6), which leads to a difference between the simulation and the actual situation. To make up for this, we propose a weak density-dependent function equation (7), which not only replaces equation (6) but also introduces electric field dependence. This improvement enables us to more accurately simulate the migration behavior of carriers, especially under high electric field conditions.

[0080] In this way, equations (7) to (10) provide an improved description of the electric field dependence. As shown in Fig. 3(b), these equations perform better in describing the electric field dependence compared to equations (3) to (6). This improvement not only enhances the prediction ability of the model but also provides a more solid theoretical basis for the design and optimization of organic electronic devices. In short, by introducing the effective temperature and considering the electric field dependence, the migration behavior of carriers in organic semiconductors can be more accurately simulated and predicted, especially under high electric field conditions. The improvement of this method is of great significance for understanding and designing organic electronic devices, providing a new perspective and tool for future development and applications.

[0081] In some embodiments, the weak density-dependent function is:

[0082]

[0083] Among them, T eff represents the effective temperature, γ is a numerical coefficient verified by theory and experiment, γ≈0.67, k B represents the Boltzmann constant, e represents the elementary charge quantity, α represents the localization length, F represents the electric field, and T represents the temperature; g(T eff , F) represents the electric field modulation function with weak density dependence, μ(T eff , p, F) represents the mobility expression improved by strengthening the electric field dependence through an exponential correction term, μ(T eff , p) represents the intrinsic mobility that only depends on the effective temperature and carrier concentration, and c1 and c2 are weak density-dependent parameters.

[0084] It can be understood that c1 and c2 significantly depend on the dimensionless quantity and the carrier concentration pb 3 .

[0085] The present invention not only solves the application limitation problem in the related art, but also calculates the carrier mobility by solving the master equation in the simulation framework. A general and concise formula is obtained to describe the charge mobility through the Marcus transition mechanism in a disordered energy environment composed of a Gaussian density of states (DOS), and its parameters are determined by accurate numerical results. The research on the small molecule TCTA organic semiconductor shows that the experimental data is highly consistent with the results fitted by the improved model. In addition, the improved mobility model uses a calibrated effective temperature to replace the actual temperature, enhancing the dependence on the electric field, so as to more accurately reflect the hopping transport law of the disordered organic semiconductor through random spatial positions. Finally, this improved method is used to analyze in detail the distribution characteristics of the charge carrier density and electric field of TCTA in the polymer layer.

[0086] The following is the experimental SCLC data analysis:

[0087] In a single-carrier device with ohmic contacts, the current is mainly limited by the internal transport in the semiconductor, and this phenomenon is usually called the space charge limited current (SCLC). In such devices, the current density (J) follows the Mott-Gurney square law:

[0088]

[0089] Where ε represents the dielectric constant of the material, V represents the applied voltage, and L represents the thickness of the layer. In this formula, the current density is proportional to the square of the applied voltage and inversely proportional to the cube of the layer thickness. In this embodiment, the hole-only device is composed of a single layer of hole-transporting material and is sandwiched between an ITO / PEDOT:PSS bottom electrode and an enhanced MoO3 / Al top electrode. For the hole-only device of TCTA, TCTA is used as the intermediate layer, and this configuration enables the direct application of the basic equations in the fields of semiconductor physics and electronic engineering to describe the charge transport and electric field distribution therein. Specifically, these equations involve the current density, the rate of change of the electric field strength, and the calculation of voltage, providing a solid theoretical basis for the in-depth understanding of the hole-transport characteristics of TCTA and its applications in electronic devices, thus providing support for the design of efficient and stable organic electronic technologies.

[0090] J = p(x)eμ(T eff , p(x), F(x))F(x) (11a);

[0091]

[0092] Here, x represents the distance from the injection electrode, p(x) refers to the hole concentration at position x in the organic film layer; ε0 is the vacuum permittivity, ε r = 3 is a typical relative permittivity value of the organic semiconductor, and L is the thickness of the polymer layer sandwiched between the two electrodes. The diffusion effect only causes a significant increase in current at low voltages, while in the region close to the electrodes, the changes in density and electric field can be ignored.

[0093] Next, an improved mobility model is used to systematically investigate the hole transport in TCTA materials. The improved mobility model is combined with the coupled equations describing the space-charge-limited current (SCLC), and the experimental J(V) measurement results of the TCTA-based hole-only device at different temperatures are also shown in Figure 4 . It should be noted that the data in the figure is presented in logarithmic scale to more clearly show the range and trend of the measured values.

[0094] Importantly, the same set of parameters is used at each temperature during simulation to ensure the consistency of the model when comparing data under different conditions. For small molecule materials, the density of states (DOS) width is usually observed to be between 0.08 - 0.15 eV. Therefore, the σ value (0.11 eV) adopted in the improved model is physically reasonable and is consistent with the known characteristics of small molecule materials.

[0095] To further verify the accuracy of the model parameters, we also use this set of parameters to describe the current-voltage characteristics of layers with different thicknesses. Figure 5The symbols in [the figure] show the J(V) characteristics of devices with different TCTA layer thicknesses (102 - 303 nm) measured at room temperature. By adjusting the thickness of the molecular layer, we found that we could accurately fit the J(V) characteristic curves of single-hole devices with different thicknesses by only assuming that the mobility μ depends on the electric field F, without changing any fitting parameters. Through this rigorous method, we ensured the effectiveness and accuracy of the model in describing hole transport in TCTA, providing a solid foundation for further research and applications.

[0096] Table 1: Comparative analysis of the energy disorder parameter (σ), lattice constant (b), mobility prefactor (μ*), and hole mobility (μ) at room temperature:

[0097]

[0098] In Table 1, we carefully compared the key parameters used in three different studies: σ (energy disorder parameter), b (lattice constant), and μ * (mobility). When exploring the hole transport characteristics in the TCTA material, we found a significant consistency between the experimental data and the improved model we proposed. This is particularly evident in the σ parameter, whose value is close to 0.112 eV used in Ref. [3] and significantly lower than 0.136 eV used in Ref. [2]. This finding reveals the accuracy of our model in describing the DOS (density of states) width of 0.1 eV, which is in good agreement with the analysis prediction based on the EGDM (effective gate dielectric constant method).

[0099] Furthermore, this similarity in energy disorder is also reflected in the mobility at room temperature. The mobility we observed is approximately in the range of 0.893×10 -8 m 2 V -1 s -1 , which is consistent with the previous research results, indicating that our improved model can accurately capture the charge transport characteristics of the material. In addition, we also noticed that the inter-site distance remains at approximately 1.4 nm in both the microscopic model and the EGDM model. This distance is equivalent to the average distance that charges jump under the action of an electric field, and this finding provides strong evidence for our understanding of the charge transport mechanism.

[0100] Finally, these results emphasize the importance of a more precise formulation of mobility. By improving the expression of mobility, we can better simulate and predict the charge transport behavior in organic semiconductor materials, which is of great significance for optimizing device performance and designing efficient electronic devices. In summary, our work not only demonstrates a good agreement between the improved model and experimental data, but also highlights the importance of accurate model parameters for understanding and predicting the behavior of organic electronic materials.

[0101] Figure 4 Shows a comparison between experimental data (symbols) and theoretical predictions (lines) of the SCL (space charge limited) current as a function of voltage for the TCTA (tris(4-carboxy-9-ylphenyl)amine) material. The symbols represent the experimental results from reference [3], while the lines depict the solution of the SCLC equation (Equation (11)) that takes into account the temperature (T), carrier concentration (p), and electric field (F) dependence of the mobility μ in the current work. The mobility parameter used in these calculations, denoted as μ * has a value of 6.1×10 -6 m 2 / Vs.

[0102] Figure 5 Shows the current density-voltage characteristics of TCTA with different thicknesses at room temperature, with symbols representing reference [3]. The figure also includes the improved EGDM simulation results, shown as lines. These simulations show excellent agreement with the experimental data over a range of different film thicknesses, using the same set of parameters cited in the main text.

[0103] Figure 6 Shows the theoretical results of the voltage as a function of charge density at the x = 0 position for a 175 nm thick TCTA material at 213 K and 295 K. The curves of different colors in the figure represent different current density J values, with the unit of A / m 3 .

[0104] Figure 7 Shows the theoretical analysis results of the voltage as a function of charge density for 303 nm and 102 nm thick TCTA layers at room temperature. The curves of different colors in the figure represent different current density J values, with the unit of A / m 3 .

[0105] Figure 6 and Figure 7Shows how the J(V) characteristics of the hole device based on TCTA change with the interface carrier density p0 at different layer thicknesses and different temperatures. It can be observed from these graphs that the voltage shows an increasing trend with the increase in current, and its change is related to p0. When p0 reaches a medium level, generally there is little correlation between the voltage and p0, which results in a flat V-p0 curve. When the carriers injected near the device surface reach an equilibrium state with the ejected carriers, the J(V) relationship enters the expected ohmic region. It is worth noting that in order to maintain the same p0 and constant current density J at low and high temperatures, the electric field enhancement and the corresponding higher voltage required at low temperature are larger than those at high temperature, indicating that the effective mobility is higher at higher temperatures. Similarly, thicker devices require a stronger electric field, while thinner devices do not; in addition, the effective mobility of thick devices is also lower than that of thin devices. Since the mobility of disordered organic materials is affected by density, very thick devices have a lower average hole density. In some cases, the effects of mobility, carrier density, and electric field enhancement can be ignored. As the device thickness decreases, the actually measured current density significantly exceeds the expected value.

[0106] Figure 8 Shows the distribution of the carrier density p along the position x for devices of different thicknesses at low and high current densities in a hole-only device based on TCTA material at room temperature. The solid lines in the figure represent the boundary carrier density p0 = 1×10 24 m 3 , and the dotted lines correspond to the case of p0 = 0.5×10 23 m 3 .

[0107] Figure 9 Shows the calculated distribution of the electric field F as a function of the position x in a hole-type device based on TCTA at room temperature, considering different thicknesses and low and high current densities. The solid lines represent the case of the boundary carrier density p0 = 0.5×10 23 m 3 , and the dotted lines represent the case of p0 = 1×10 24 m 3 .

[0108] In Figure 8 and Figure 9 , the effects of position (distance from the interface) on the carrier density and electric field in a TCTA-based hole-only device are analyzed in detail. We must pay special attention to the mobility in the central region of the device, where p0 is approximately 10 23 -10 24 m -3 , to ensure that the J(V) characteristics are physically realistic. Therefore, we used two different p0 values in the calculation, namely [(0.5, 10)×10 23 m-3 In the case of disordered materials, there are no explicit equations for the carrier density p(x) and the electric field F(x). The numerical results show that p(x) decreases gradually with the increase of the distance x, while the distribution of F(x) increases with the increase of the distance x from the ITO injection anode. For larger values of p0, the rate of change is greater; while for smaller values of p0, the rate of change is smaller. The Poisson equation (Equation (11b)) shows that the electric field increases monotonically with the change of position and relates the electric field to the carrier density. Therefore, the carrier injection from the electrode to the TCTA layer results in an enhancement of the space charge near the interface and a decreasing trend of p(x). The distribution of p(x) and the enhancement of the space charge near the interface cause the change of F(x). These findings are similar to the phenomena of carrier density and field-dependent mobility described by Tanase et al.

[28] in a drift system.

[0109] The relevant references mentioned above are as follows:

[0110] [1] Pawlik, W. F., Kotar, J., Tanase, C., et al. "Unified description of charge carrier mobility in disordered semiconducting polymers" [J]. Physical Review Letters, 2005, 94(20): 206601.

[0111] [2] Masse, A., Friedrich, P., Cimalla, F., et al. "First-principles-based model for charge carrier mobility in amorphous molecular semiconductors" [J]. Physical Review B, 2016, 93(19): 195209.

[0112] [3] Kotadia, N. B., Anirban, M., Xiong, S., et al. "Rigorous characterization and predictive modeling of hole transport in amorphous organic semiconductors" [J]. Advanced Electronic Materials, 2018: 1800366.

[0113] Corresponding to Figure 1 the method of Figure 10 , an embodiment of the present invention provides a system for predicting the mobility of semiconductor carriers, including:

[0114] At least one processor;

[0115] At least one memory for storing at least one program;

[0116] When the at least one program is executed by the at least one processor, the at least one processor implements the above method.

[0117] It can be seen that the content in the above method embodiments is applicable to the system embodiments of the present invention. The functions specifically implemented by the system embodiments of the present invention are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those of the above method embodiments.

[0118] In addition, an embodiment of the present invention also discloses a computer program product or a computer program, which is stored in a computer-readable storage medium. The processor of the computer device can read the computer program from the computer-readable storage medium, and the processor executes the computer program, so that the computer device executes the above-mentioned method. Similarly, the content in the above method embodiments is applicable to this storage medium embodiment. The functions specifically implemented by this storage medium embodiment are the same as those in the above method embodiments, and the beneficial effects achieved are also the same as those in the above method embodiments.

[0119] Those of ordinary skill in the art can understand that all or some of the methods and systems disclosed above can be implemented as software, firmware, hardware, and their appropriate combinations. Some physical components or all physical components can be implemented as software executed by a processor, such as a central processing unit, a digital signal processor, or a microprocessor, or can be implemented as hardware, or can be implemented as an integrated circuit, such as an application-specific integrated circuit. Such software can be distributed on a computer-readable medium, which can include a computer storage medium (or non-transitory medium) and a communication medium (or transitory medium). As is well known to those of ordinary skill in the art, the term computer storage medium includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storing information, such as computer-readable instructions, data structures, program modules, or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory, or other memory technologies, CD-ROM, digital versatile disk (DVD), or other optical disk storage, magnetic cassette, tape, magnetic disk storage, or other magnetic storage devices, or any other medium that can be used to store the desired information and can be accessed by a computer. In addition, as is well known to those of ordinary skill in the art, a communication medium typically includes computer-readable instructions, data structures, program modules, or other data in a modulated data signal such as a carrier wave or other transmission mechanism, and can include any information delivery medium.

[0120] The above is a specific description of the preferred embodiments of the present disclosure, but the present disclosure is not limited to the above embodiments. Those skilled in the art can also make various equivalent deformations or substitutions without departing from the spirit of the present disclosure, and these equivalent deformations or substitutions are all included within the scope defined by the claims of the present disclosure.

Claims

1. A method for predicting the mobility of semiconductor carriers, characterized in that, The method includes the following steps: Establish the Marcus hopping rate that characterizes the transport of charge through a discontinuous hopping mechanism in an organic semiconductor; Based on the Marcus hopping rate, establish a master equation under steady-state conditions, which is used to characterize that the charge occupancy probabilities of each site in the simulated charge transport system are interdependent through the charge hopping process and reach equilibrium under steady state; Establish a hole mobility model and a density-independent mobility correction factor, and establish a mobility model based on the hole mobility model and the mobility correction factor; wherein, the mobility correction factor is used to characterize the electric field dependence, and the hole mobility model is used to characterize the relationship between the hole mobility and the carrier concentration at different temperatures under zero electric field conditions; Establish a weak density-dependent function based on the effective temperature, establish an improved mobility model based on the weak density-dependent function and the mobility model, and predict the mobility of semiconductor carriers based on the improved mobility model; wherein, the effective temperature is used to characterize the temperature experienced by the carriers under the action of an electric field.

2. The method according to claim 1, wherein The formula for the Marcus hopping rate is as follows: Among them, W ij represents the hopping rate, represents the reduced Planck constant, k B represents the Boltzmann constant, T represents the temperature, E r is a parameter representing the molecular reorganization energy, ΔE ij = E i - E j defines the energy difference between sites i and j, J ij represents the charge transfer integral between these two sites.

3. The method according to claim 2, wherein The master equation under steady-state conditions is expressed as: Among them, p i represents the probability that site i is occupied by a charge, and p j represents the probability that site j is occupied by a charge. W ij is the hopping rate of the charge from site i to site j, and W ji is the hopping rate of the charge from site j to site i.

4. The method according to claim 3, wherein The calculation formula for the hole mobility model is: μ(T,p,F)≈μ(T,p)f(T,F)(3); Among them, μ(T,p,F) represents the hole mobility before improvement, μ(T,p) represents the intrinsic mobility that only depends on temperature and carrier concentration, f(T,F) is the mobility correction factor at electric field F and temperature T, μ0(T) represents the dimensionless scaling factor of the intrinsic mobility, which is used to adjust the mobility baseline value at zero carrier concentration (p→0), and μ * represents the scaled intrinsic mobility, c = 0.42, which represents the exponential decay coefficient of the temperature dependence of the mobility; represents the temperature-normalized energy disorder, p represents the carrier concentration, b represents the lattice constant, and δ represents the exponential adjustment factor of the influence of carrier density on the mobility.

5. The method according to claim 4, wherein The calculation formula for the mobility correction factor is: where f(T,F) represents the mobility correction factor at an electric field F and temperature T, represents the temperature-normalized energy disorder, σ represents the energy disorder, ebF represents the energy modulation effect of the electric field on carrier migration, and A and B represent empirical fitting parameter factors.

6. The method according to claim 4, wherein The weak density-dependent function is: Among them, T eff represents the effective temperature, γ is a numerical coefficient, k B represents the Boltzmann constant, e represents the elementary charge quantity, α represents the localization length, F represents the electric field, and T represents the temperature; g(T eff , F) represents the electric field modulation function with weak density dependence, μ(T eff , p, F) represents the mobility expression improved by strengthening the electric field dependence through an exponential correction term, μ(T eff , p) represents the intrinsic mobility that only depends on the effective temperature and the carrier concentration, and c1 and c2 are weak density-dependent parameters.

7. A prediction system for the mobility of semiconductor carriers, characterized in that, The system includes: At least one processor; At least one memory for storing at least one program; When the at least one program is executed by the at least one processor, the at least one processor implements the method according to any one of claims 1 to 6.

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