Method based on plasma shock wave propagation in quantum field effect transistor

Abstract

The application discloses a propagation model of plasma shock wave in a quantum field effect transistor, and is based on a self-consistent quantum hydrodynamic model for describing characteristics of a two-dimensional electron gas in a conductive channel of a field effect transistor, adopts a reduced perturbation method to expand the model, obtains a KdV-Burgers model for describing propagation characteristics of the plasma shock wave in the conductive channel of the field effect transistor, and solves a numerical solution of the model, and the numerical simulation result gives the distribution of the plasma shock wave with time and space. The dynamics analysis further verifies the model. The characteristics of the propagation of the plasma shock wave in the quantum field effect transistor are discussed. The model greatly reduces the complexity of calculation.

Description

Technical Field

[0001] This invention belongs to the field of semiconductor quantum device technology, specifically relating to a method for plasma shock wave propagation in a quantum field-effect transistor. Background Technology

[0002] Since the beginning of the 21st century, the semiconductor industry has experienced rapid development, with field-effect transistors (FETs) reaching nanometer-scale dimensions. At this point, because the geometric dimensions of FETs are comparable to the wavelength of electrons, the impact of quantum effects on device performance cannot be ignored. Taking N-type semiconductors as an example, at the microscopic scale, the movement of electrons is confined within quantum wells. Each electron possesses a wave function that conforms to the Schrödinger equation; therefore, the energy of the electrons is quantized, and the energy levels near the Fermi level change from quasi-continuous to discrete. During device miniaturization, electrons are modulated by quantum effects, leading to significant differences in the operating behavior of electrons in FETs compared to traditional FETs.

[0003] Shock waves are a crucial type of nonlinear coherent structure with wide applications in chemistry, medicine, optics, aerospace, and astrophysics. In fact, shock wave structures also exist in field-effect transistors (FETs). When plasma flows from the source to the drain in the conductive channel of an FET, it is reflected and amplified at the drain, causing the original plasma to become unstable. This results in a non-uniform distribution of plasma density in the channel, forming a plasma shock wave. Plasma shock waves can be further classified into oscillating plasma shock waves and monotonic plasma shock waves. This undoubtedly lays a solid theoretical foundation for my country to realize usable and tunable high-power semiconductor devices.

[0004] Currently, on the one hand, researchers are studying quantum effects in field-effect transistors by solving the Schrödinger equation. However, in practical problems, the analytical solution to the Schrödinger equation is difficult to obtain, and the numerical solution only holds under finite conditions. On the other hand, solving fluid dynamics models requires not only repeated iterations but also adherence to relevant fluid dynamics principles and mathematical convergence conditions. Crucially, the solution to this model is not unique, resulting in high difficulty and computational complexity. Summary of the Invention

[0005] To address the aforementioned problems of high difficulty and computational complexity in solving the equation, this invention primarily provides a propagation model for plasma shock waves in quantum field-effect transistors. By using the KdV-Burgers model, the difficulty in solving the Schrödinger equation is avoided. Furthermore, it avoids the iterative processes involved in directly solving the Schrödinger equation and solving conventional quantum hydrodynamic models. The invention focuses on the shock waves generated in the system when dispersion caused by quantum effects and dissipation generated by electron fluid dynamics interact.

[0006] The technical solution adopted in this invention is as follows:

[0007] The method for plasma shock wave propagation in quantum field-effect transistors includes the following steps:

[0008] Step 1: Propose a model and use the reduction perturbation method to expand the quantum hydrodynamic model, expanding all variables in the model except for time and space into polynomials with perturbations;

[0009] Step 2: Propose a model by arranging and merging the perturbation quantities from low to high powers to obtain the KdV-Burgers model that describes the propagation characteristics of plasma shock waves in the conductive channel of a field-effect transistor.

[0010] Step 3: Calculate the model, provide the initial and boundary conditions for solving the KdV-Burgers model, and obtain the numerical solution of the model;

[0011] Step 4: Calculate the model. Use the finite difference method and Newton's iteration method to solve the numerical solution of the model proposed in Step 2. Use the controlled variable method to obtain the spatiotemporal distribution map of the plasma shock wave.

[0012] Step 5: Verify the model. Perform dynamic analysis on the KdV-Burgers model and use the traveling wave transform method to convert the KdV-Burgers model into a system of ordinary differential equations.

[0013] Step 6: Verify the model, provide the initial and boundary conditions for solving the system of ordinary differential equations, and obtain the phase trajectory of the system as a function of parameters. This phase trajectory is consistent with the spatiotemporal distribution of the plasma shock obtained in Step 4. Figure 1 One-to-one correspondence.

[0014] The specific method for step 1 is as follows:

[0015] The propagation of plasma shock waves in a field-effect transistor can be described using a quantum hydrodynamic model as follows:

[0016]

[0017] (1)

[0018]

[0019] The reduction perturbation method is used to replace each variable in the quantum hydrodynamic model with perturbations, that is, to substitute equations (2) and (3) into equation (1).

[0020] (2)

[0021] (3)

[0022] The KdV-Burgers model for the propagation characteristics of plasma shock waves in the conductive channel of the field-effect transistor obtained in step 2 is as follows:

[0023] (4)

[0024] This model is a typical third-order partial differential equation, in which... H is a viscosity coefficient derived from electron fluidity, used to characterize dissipation, while H is a quantum effect coefficient derived from quantum effects, used to characterize the quantum effect coefficient of dispersion. and These are time coordinates and spatial coordinates, respectively. Essentially, it represents plasma density or the power of plasma radiating outwards.

[0025] The specific method for step 5 is as follows:

[0026] Using the traveling wave transform method The model proposed in step 2 is transformed into a system of ordinary differential equations, namely (5). The correctness of step 4 is verified from the perspective of nonlinear dynamics, that is, the numerical solution of the system of ordinary differential equations is obtained.

[0027] (5)

[0028] The specific method for step 6 is as follows:

[0029] After providing specific initial and boundary conditions, the numerical solution of the ordinary differential equation system corresponding to equation (5) is obtained. To more accurately and vividly illustrate this process, the parameters are... As the values ​​gradually change from small to large, we present the phase trajectories of the system under several key parameters. These phase trajectories are related to the spatiotemporal distribution of the plasma shock wave calculated earlier. Figure 1 The one-to-one correspondence further verifies the accuracy of the model proposed in this patent.

[0030] The propagation model of plasma shock waves in a quantum field-effect transistor includes a model proposal module, a computational model module, and a verification model module. The model proposal module is used to derive the KdV-Burgers model proposed in this patent from traditionally difficult-to-solve quantum hydrodynamic systems, greatly simplifying the computational complexity. The computational model module is used to directly solve the KdV-Burgers model, providing a spatiotemporal distribution diagram of the plasma shock waves in the quantum field-effect transistor. The verification model module is used to verify the correctness of the proposed model, providing the phase trajectory of the system from a nonlinear dynamics perspective, further verifying the accuracy of the proposed model.

[0031] Compared with the prior art, the beneficial effects of the present invention are:

[0032] This invention proposes a propagation model for plasma shock waves in a quantum field-effect transistor (QFET), comprising a model proposal module, a computational model module, and a verification model module. The model proposal module derives the proposed KdV-Burgers model from traditionally difficult-to-solve quantum hydrodynamic systems, greatly simplifying computational complexity. The computational model module directly solves the KdV-Burgers model, providing a spatiotemporal distribution diagram of the plasma shock wave in the QFET. The verification model module verifies the correctness of the proposed model, providing the phase trajectory of the system from a nonlinear dynamics perspective, further validating the accuracy of the proposed model.

[0033] This invention uses the KdV-Burgers model, circumventing the difficulty in solving the Schrödinger equation. It also avoids the iterative processes involved in directly solving the Schrödinger equation and solving conventional quantum hydrodynamic models. The invention focuses on the shock waves generated in the system when dispersion caused by quantum effects and dissipation generated by electron fluid dynamics work together. Attached Figure Description

[0034] Figure 1 This is a flowchart of the plasma shock wave propagation model in the quantum field-effect transistor of this invention.

[0035] Figure 2 The image shows the propagation model of plasma shock waves in the quantum field-effect transistor of this invention, along with shock wave waveforms under different viscosity coefficients.

[0036] Figure 3 This is a diagram showing the evolution of the plasma shock wave over time and space under different quantum effects in the quantum field-effect transistor of this invention.

[0037] Figure 4 This is the phase trajectory of the plasma shock wave propagation model in the quantum field-effect transistor of this invention. Detailed Implementation

[0038] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are implemented based on the technical solution of the present invention, providing detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0039] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Of course, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0040] This invention avoids the Schrödinger equation in its solution process and proposes a new model based on quantum hydrodynamics to simulate plasma shock waves in field-effect transistors. Its basic framework is as follows: Figure 1 As shown, this invention includes a model proposal module, a computational model module, and a verification model module. The model proposal module is used to derive the KdV-Burgers model proposed in this patent from traditionally difficult-to-solve quantum hydrodynamic systems, greatly simplifying the computational complexity. The computational model module is used to directly solve the KdV-Burgers model, providing a spatiotemporal distribution diagram of the plasma shock wave in the quantum field-effect transistor. The verification model module is used to verify the correctness of the proposed model, providing the phase trajectory of the system from the perspective of nonlinear dynamics, further verifying the accuracy of the proposed model.

[0041] This invention provides a method for plasma shock wave propagation in a quantum field-effect transistor, comprising the following steps:

[0042] Step 1: Propose a model and use the reduction perturbation method to expand the quantum hydrodynamic model, expanding all variables in the model except for time and space into polynomials containing perturbations.

[0043] The propagation of plasma shock waves in a field-effect transistor is described using a quantum hydrodynamic model as follows:

[0044]

[0045] (1)

[0046]

[0047] The reduction perturbation method is used to replace each variable in the quantum fluid dynamics model with perturbations, that is, to substitute equations (2) and (3) into equation (1).

[0048] (2)

[0049] (3)

[0050] Step 2: Propose a model by merging the perturbation quantities from low to high powers to obtain the KdV-Burgers model that describes the propagation characteristics of plasma shock waves in the conductive channel of a field-effect transistor.

[0051] (4)

[0052] This model is a typical third-order partial differential equation, in which... H is the viscosity coefficient derived from electron fluidity and used to characterize dissipation, while H is the quantum effect coefficient derived from quantum effects and used to characterize dispersion. and These are time coordinates and spatial coordinates, respectively. Essentially, it represents plasma density or the power of plasma radiating outwards.

[0053] Step 3: Calculate the model, providing the initial and boundary conditions for solving the KdV-Burgers model. Because the solutions to partial differential equations vary greatly depending on the initial and boundary conditions, and may even be unsolvable, accurate, appropriate initial and boundary conditions consistent with the physical model must be provided. The numerical solution of the model is then obtained.

[0054] Step 4: Calculate the model. Use the finite difference method and Newton's iteration method to solve the numerical solution of the model proposed in Step 2. Use the controlled variable method to obtain the spatiotemporal distribution map of the plasma shock wave.

[0055] Figure 2 Shock wave waveforms for different viscosity coefficients are presented. From Figure 2 As can be seen, when the viscosity coefficient is small, the oscillation of the plasma shock wave in the quantum field-effect transistor is obvious. When the viscosity coefficient increases, the oscillation of the shock wave weakens, and when the viscosity coefficient reaches 0.03, the oscillation of the shock wave eventually disappears. This indicates that when the viscosity coefficient reaches a certain value, there is a transformation from oscillating shock wave to monotonic shock wave in the system.

[0056] Figure 3 The evolution of shock waves over time and space under different quantum effects is presented, from... Figure 3 It is easy to see that as the quantum effect intensifies, the shock wave oscillations become increasingly intense.

[0057] Figure 2 and Figure 3Together, these findings demonstrate the existence of two types of shock waves in the conductive channel of a quantum field-effect transistor: oscillating and monotonic. These two types can transform into each other under certain conditions. Numerical calculations show that viscosity weakens the shock wave oscillations, and the oscillations become increasingly intense with the increase of quantum effects. Furthermore, in a system where viscosity and quantum effects work together, viscosity always leads the shock wave towards a monotonic form, while quantum effects always lead it towards an oscillating form.

[0058] Step 5: Verify the model using the traveling wave transform method. The model proposed in step 2 is transformed into a system of ordinary differential equations, namely (5). The correctness of step 4 is verified from the perspective of nonlinear dynamics, that is, the numerical solution of the system of ordinary differential equations is obtained.

[0059] (5)

[0060] Step 6: Verify the model. After providing the initial and boundary conditions for solving the system of ordinary differential equations, the numerical solution of the system of ordinary differential equations corresponding to equation (5) is obtained. To more accurately and vividly illustrate this process, let the parameters... As the values ​​gradually change from small to large, we present the phase trajectories of the system under several key parameters. These phase trajectories are related to the spatiotemporal distribution of the plasma shock wave calculated earlier. Figure 1 The one-to-one correspondence further verifies the accuracy of the model proposed in this patent.

[0061] Figure 4 The KdV-Burgers model proposed in step two was further validated from the perspective of nonlinear dynamics. Figure 4 It can be seen that the solution to the ordinary differential equations proposed in step five is a series of phase trajectories. With the increase of the viscosity coefficient, the heteroclinic trajectory connecting the saddle and focus transforms into a heteroclinic trajectory connecting the saddle and node. The heteroclinic trajectory connecting the saddle and focus corresponds to the oscillating shock wave solution of the model proposed in step two, i.e. Figure 4 (a)(b) and Figure 2 Corresponding to (a) and (b) in the above, in this case, the dispersion produced by quantum effects in the system is greater than the dissipation produced by the electron fluid. Similarly, the saddle-nodal heteroclimate orbit in the phase trajectory corresponds to the monotonic shock wave solution of the model proposed in step two, i.e. Figure 4 (c) and Figure 2 Corresponding to (c) in the model, the dissipation generated by the electron fluid in the system is greater than the dispersion produced by quantum effects. The oscillating shock wave is transformed into a monotonic shock wave, thus verifying the correctness of Model 2.

[0062] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for plasma shock wave propagation in a quantum field-effect transistor, characterized in that, Includes the following steps: Step 1: Propose a model and use the reduction perturbation method to expand the quantum hydrodynamic model, expanding all variables in the model except for time and space into polynomials with perturbations; Step 2: Propose a model by arranging and merging the perturbation quantities from low to high powers to obtain the KdV-Burgers model that describes the propagation characteristics of plasma shock waves in the conductive channel of a field-effect transistor. Step 3: Calculate the model, provide the initial and boundary conditions for solving the KdV-Burgers model, and obtain the numerical solution of the model; Step 4: Calculate the model. Use the finite difference method and Newton's iteration method to solve the numerical solution of the model proposed in Step 2. Use the controlled variable method to obtain the spatiotemporal distribution map of the plasma shock wave. Step 5: Verify the model. Perform dynamic analysis on the KdV-Burgers model and use the traveling wave transform method to convert the KdV-Burgers model into a system of ordinary differential equations. Step 6: Verify the model, provide the initial and boundary conditions for solving the system of ordinary differential equations, and obtain the phase trajectory of the system as the parameters evolve; this phase trajectory corresponds one-to-one with the spatiotemporal distribution diagram of the plasma shock wave obtained in Step 4.

2. The method for plasma shock wave propagation in a quantum field-effect transistor according to claim 1, characterized in that, The specific method for step 1 is as follows: The propagation of plasma shock waves in a field-effect transistor can be described using a quantum hydrodynamic model as follows: （1） The reduction perturbation method is used to replace each variable in the quantum hydrodynamic model with perturbations, that is, to substitute equations (2) and (3) into equation (1). (2) (3) 。 3. The method for plasma shock wave propagation in a quantum field-effect transistor according to claim 1, characterized in that, The KdV-Burgers model for the propagation characteristics of plasma shock waves in the conductive channel of the field-effect transistor obtained in step 2 is as follows: (4) In the formula H is the viscosity coefficient used to characterize dissipation, while H is the quantum effect coefficient used to characterize dispersion. and These are time coordinates and spatial coordinates, It represents the plasma density or the power radiated outward by the plasma.

4. The method for plasma shock wave propagation in a quantum field-effect transistor according to claim 1, characterized in that, The specific method for step 5 is as follows: Using the traveling wave transform method The model proposed in step 2 is transformed into a system of ordinary differential equations, namely (5); (5) To verify the correctness of step 4 from the perspective of nonlinear dynamics, we need to find the numerical solution of the system of ordinary differential equations.

5. The method for plasma shock wave propagation in a quantum field-effect transistor according to claim 1, characterized in that, The specific method for step 6 is as follows: After providing specific initial and boundary conditions, the numerical solution of the ordinary differential equation system corresponding to equation (5) is obtained. To more accurately and vividly illustrate this process, the parameters are... The phase trajectories of the system are given under several key parameters, gradually changing from small to large. These phase trajectories correspond one-to-one with the spatiotemporal distribution diagram of the plasma shock wave obtained earlier, thereby further verifying the accuracy of the model.

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