A method to investigate the electrical transport properties of Al / α-Al2O3 / Al by changing the thickness of the barrier layer based on COMSOL

Abstract

This invention discloses a method for investigating the electrical transport properties of Al / α-Al2O3 / Al by varying the barrier layer thickness using COMSOL. The method comprises: utilizing the Schrödinger interface of the COMSOL semiconductor module simulation software to construct a one-dimensional square barrier tunneling model and a three-dimensional Al / α-Al2O3 / Al Josephson junction device model; and performing a parametric sweep of the barrier layer thicknesses for both models to determine the quantum tunneling effect of particles and the effect of the aluminum oxide film thickness on the electrical transport properties of the Josephson junction device. This method provides a direction for improving existing Josephson junction fabrication processes and verifies the feasibility and effectiveness of studying Josephson junctions based on a multi-physics field coupling model, which is of great significance for improving the performance of Josephson junctions and superconducting integrated circuits.

Description

Technical Field

[0001] The present invention relates to the field of superconducting integrated circuit technology in the core electronics industry, and in particular to a method for exploring the electrical transport properties of Al / α-Al2O3 / Al by changing the thickness of a barrier layer based on COMSOL. Background Art

[0002] The rapid development of superconducting quantum computers is largely due to the improvement of their decoherence time, such as improved circuit design, the use of new materials and improved manufacturing processes. Research focusing on new materials is mostly in the selection of materials for capacitors or microwave resonators, while there is little exploration of Josephson junction materials. Advanced computational modeling theories, such as first principles, can provide a faster way to optimize processes and improve performance. They can quickly apply calculation results to process preparation, save costs, speed up experimental iteration cycles, and improve process manufacturing. At present, research based on the atomic scale of Josephson junctions has achieved certain results, and it has been found that changing the atomic arrangement can improve the performance of Josephson junctions to a certain extent. However, research on the electrical properties of Josephson junctions at the mesoscopic scale based on the COMSOL multi-physics field coupling finite element method is still seriously lacking. Summary of the Invention

[0003] In order to better and more intuitively understand the impact of changes in the mesoscopic scale of the Josephson junction on its electrical transport properties, and to prove that the simulation of key components of quantum chips using a multi-physics coupling model is effective, this paper proposes a method for exploring the electrical transport properties of Al / α-Al2O3 / Al by changing the thickness of the barrier layer based on COMSOL. Using the Schrödinger interface in the COMSOL semiconductor module, a one-dimensional square barrier tunneling model is first built to explore the quantum tunneling effect, and an equation for calculating the particle transmission coefficient is customized in COMSOL. Then, a three-dimensional Josephson junction nm-scale simulation model is built, and the transmission coefficient formula for calculating the one-dimensional model is derived and imitated to obtain a formula for calculating the particle tunneling probability of the three-dimensional model in the multi-physics coupling model. Finally, this formula is combined with the Ambegaokar-Baratoff relationship to obtain the relevant electrical transport results of the Josephson junction, and the influence of the thickness of the Josephson junction barrier layer on its electrical properties is explored. The present invention provides a direction for improving the existing Josephson junction preparation process and verifies the feasibility and effectiveness of studying Josephson junctions based on a multi-physics field coupling model, which is of great significance for improving the performance of Josephson junctions and superconducting integrated circuits.

[0004] In order to achieve the above object, the present invention adopts the following technical solutions:

[0005] A COMSOL-based method for investigating the electrical transport properties of Al / α-Al2O3 / Al by varying the barrier layer thickness includes:

[0006] Step 1: Use the Schrödinger interface of the COMSOL Semiconductor Module to build a one-dimensional square barrier tunneling model and a three-dimensional Al / α-Al2O3 / Al Josephson junction device model.

[0007] Step 2: Perform parameterized scans on the barrier layer thickness of the two models to obtain the quantum tunneling effect of particles and the influence of the thickness of the aluminum oxide film on the electrical transport properties of the Josephson junction device.

[0008] Furthermore, the step 2 includes:

[0009] Step 2.1: Calculate the transmission coefficient of the one-dimensional square barrier tunneling model;

[0010] Step 2.2: Calculate the tunneling probability of particles passing through barrier layers of different thicknesses in the 3D Al / α-Al2O3 / Al Josephson junction device model;

[0011] Step 2.3: Substitute the tunneling probability into the Ambegaokar-Baratoff relationship to obtain the critical current density and room temperature resistance of the Josephson junction device model.

[0012] Furthermore, in step 2.1, the transmission coefficient of the one-dimensional square barrier tunneling model is calculated as follows:

[0013]

[0014] in Represents the transmission coefficient of the one-dimensional square barrier tunneling model, psi0 is the custom incident plane wave amplitude, schr.open1.int(psi) represents the wave function amplitude calculated at the open boundary condition named open1, open1 represents the rightmost endpoint of the one-dimensional square barrier tunneling model, and abs() represents taking the absolute value.

[0015] Furthermore, in step 2.2, the tunneling probability of particles passing through the barrier layer in the three-dimensional Al / α-Al2O3 / Al Josephson junction device model is calculated as follows:

[0016] ξ=abs(aveop(schr.Pr_psi) / psi0)^2 (5)

[0017] Where aveop() is the average operator defined in the physical field nonlocal coupling, aveop(schr.Pr_psi) represents the average probability density at the cross-sectional area of the top aluminum electrode, abs() represents the absolute value, and psi0 is the custom incident plane wave amplitude.

[0018] Furthermore, in step 2.3, the critical current density J of the Josephson junction device model is calculated as follows: S and the constant temperature resistance R n :

[0019]

[0020] R n =2πh / (e 2 ξ×DOS) (7)

[0021] in ξ is the tunneling probability, DOS is the density of states at the Fermi level, Δ(T) represents the superconducting energy gap in volts, T represents the temperature, e is the charge, h is the Planck constant, and k B is the Boltzmann constant.

[0022] Compared with the prior art, the present invention has the following beneficial effects:

[0023] In the present invention, we characterized the quantum tunneling effect of the one-dimensional square barrier tunneling model and the related electrical transport properties of the three-dimensional Josephson junction. Based on the multi-physics field coupling model method, the barrier layer thickness was parametrically studied and calculated, and it was found that the barrier layer thickness can affect the transmission coefficient of the particle and thus affect the conductive properties of the device. In the one-dimensional model, the wave function of the particle was calculated based on COMSOL to solve the Schrödinger equation. It was found that the wave function amplitudes on the left and right sides of the barrier layer were different, which proved that even if the energy of the particle is lower than the energy of the barrier region, it has a probability of passing through the barrier region and losing some energy, that is, the particle will undergo both transmission and reflection, which is the quantum tunneling effect. In the three-dimensional model, by post-processing the calculation results, a custom formula for the tunneling probability of the particle passing through the three-dimensional Josephson junction was solved, and the formula was combined with the Ambegaokar-Baratoff relationship to obtain the critical current density and room temperature resistance of the Josephson junction. By varying the thickness of the intermediate insulating film barrier layer, the authors found that the tunneling probability of the particles fluctuated, and that their room-temperature resistance increased exponentially with increasing barrier layer thickness. These conclusions are consistent with those obtained from related physical experimental studies, demonstrating the feasibility of studying Josephson junction devices using a multi-physics coupling model. This lays an important technical foundation for future characterization and simulation of superconducting quantum chips and circuits based on multi-physics coupling models. This paper uses simulation to study complex Josephson junction devices at the mesoscopic scale, saving research costs and accelerating the research process, laying the foundation for further in-depth exploration of Josephson junction devices. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1This is a flow chart of a method for exploring the electrical transport properties of Al / α-Al2O3 / Al by changing the thickness of the barrier layer based on COMSOL according to an embodiment of the present invention;

[0025] Figure 2 A one-dimensional square barrier potential energy function diagram provided by an embodiment of the present invention;

[0026] Figure 3 A schematic diagram of a Josephson junction device model constructed according to an embodiment of the present invention;

[0027] Figure 4 Summary diagram of potential energy, energy, and wave function provided by embodiments of the present invention;

[0028] Figure 5 A curve showing the variation of particle tunneling probability with the thickness of the barrier layer provided in an embodiment of the present invention;

[0029] Figure 6 The curve of the Josephson junction temperature resistance changing with the barrier thickness provided by the embodiment of the present invention. DETAILED DESCRIPTION

[0030] The present invention will be further explained below with reference to the accompanying drawings and specific embodiments:

[0031] like Figure 1 As shown in Figure 1, a COMSOL-based method for investigating the electrical transport properties of Al / α-Al2O3 / Al by changing the barrier layer thickness includes:

[0032] 1. Using the Schrödinger interface of the COMSOL semiconductor module, we built a one-dimensional square barrier tunneling model and a three-dimensional Al / α-Al2O3 / Al Josephson junction device model.

[0033] (1) Geometric modeling: Build a one-dimensional square barrier tunneling model, where the width of the potential well is 15nm, the width of the barrier is 1nm, and the width of the modeling domain ranges from 30.5nm to 32.5nm. Build a three-dimensional rectangular Josephson junction device model, where the cross-sectional area of the Josephson junction device is 10nm*10nm, the width of the superconducting metal aluminum electrodes on both sides is 15nm, and the thickness of the Al2O3 insulating film in the middle ranges from 1nm to 2.5nm.

[0034] (2) Material definition: In the three-dimensional Josephson junction device model, the electrode materials on both sides are the metal aluminum defined in the COMSOL built-in material library, and the middle oxide layer film is defined as the insulator Al2O3.

[0035] (3) Schrödinger interface parameter settings: The steady-state study energy E represents the total energy required to solve the Schrödinger equation, which refers to the energy of the tunneling particle; the effective mass of the electron in the entire domain is 0.067 times the mass of the free electron; the electronic potential energy of different materials is calculated based on DFT (density-functional theory); open boundary conditions and periodic boundary conditions are defined in the three-dimensional model.

[0036] (4) Meshing: Both the one-dimensional square barrier tunneling model and the three-dimensional Josephson junction device use user-defined meshing. The overall idea of meshing is that in one dimension, the number of meshes in the potential well region is sparser than that in the barrier region; in three dimensions, a hexahedral mesh is constructed using a sweeping method. Similarly, the mesh distribution in the barrier film is denser. The average mesh quality of this method is around 0.7.

[0037] 2. Research and calculate the built model

[0038] (1) Based on the selected semiconductor physics and interface parameter settings, COMSOL uses finite element methods to solve the Schrödinger equation. The defined meshing method determines the calculation time and result accuracy of the study.

[0039] (2) Parametric scanning: The parametric scanning method is used for both constructed models, and the barrier width and aluminum oxide film thickness in the one-dimensional square barrier tunneling model and the three-dimensional Josephson junction device are changed respectively. The initial value is set to 0.5nm, the change step is 0.01nm, and the end value is 2.5nm.

[0040] (3) In the COMSOL Schrödinger interface, the default solution obtained includes: the particle wave function image, probability density, potential energy and effective mass of the model.

[0041] 3. Post-processing of results

[0042] Using the default solution provided by COMSOL, the results were post-processed using built-in COMSOL functions to obtain the transmission coefficient of a particle through a one-dimensional square potential barrier, the tunneling probability of a particle (Cooper pair) through a three-dimensional Josephson junction device, the critical current density, and the room-temperature resistance of the Josephson junction device. Comparison of the wave function changes for particles passing through the one-dimensional square potential barrier model with varying barrier thicknesses confirmed the quantum tunneling effect of the particle. The effect of the thickness of the aluminum oxide insulating film on the electrical transport properties (critical current density and room-temperature resistance) of the Josephson junction device in the three-dimensional model was discussed.

[0043] The method specifically includes:

[0044] 1. Model building

[0045] (1) One-dimensional square barrier tunneling model

[0046] To verify that COMSOL can calculate the tunneling probability of particles passing through a three-dimensional Josephson junction device, we first built a one-dimensional square barrier tunneling model in COMSOL. The potential barrier is a distribution form of a potential energy function, and its functional expression can be described as:

[0047]

[0048] Among them E P (x) represents the potential energy at the x-coordinate, a represents a constant on the x-axis, E p0 Represents a certain energy value when energy is used as the vertical axis.

[0049] We used numerical simulation in COMSOL to build Figure 2 The modeling domain shown is a 31nm one-dimensional square potential barrier model. The one-dimensional model is constructed using line segment spacing. The model specifies the spacing between the potential well and barrier regions starting at -15nm in the coordinate system. The horizontal axis represents the x-coordinate of the model in one-dimensional space in nm, and the vertical axis represents the potential energy in eV. To set and measure the wave function, open boundary conditions are set at both the left and right endpoints of the model.

[0050] (2) Three-dimensional Josephson junction device model

[0051] In order to calculate the tunneling probability of particles passing through the Josephson junction device and characterize the electrical transport properties of the device, we used COMSOL to build a three-dimensional Al / α-Al2O3 / Al Josephson junction device model, as shown in Figure 3 The 3D model is created by first constructing a 2D work plane and then stretching it. The model is then layered in the z-direction, dividing a solid rectangle into three layers, one for the bottom aluminum electrode, the aluminum oxide insulating film, and the top aluminum electrode. To reduce simulation time, the actual junction cross-sectional area is reduced to 10nm*10nm, the width of the aluminum metal on both sides is 15nm, and the initial width of the middle insulating film is defined as 1nm.

[0052] To solve the Schrödinger equation in the three-dimensional model, a steady-state study method is used. The steady-state study energy E is defined as 11.684 eV, which is the energy at the Fermi level of aluminum and represents the energy of the Cooper pairs in bulk aluminum. The electron potential energy of the aluminum oxide in the middle and the metal regions on both sides are calculated based on DFT. The effective mass of the electron in the entire modeling domain is defined as m eff= m*0.067, where m is the mass of an electron. The device has a periodic structure in the x and y directions, with the z direction being the device's electron transport direction. Open boundary conditions are set at the bottom and top aluminum electrodes. The open boundary condition at the bottom aluminum electrode is used to set the amplitude of the incident wave, while the open boundary condition at the top aluminum electrode is used to measure the outgoing wave.

[0053] 2. Model calculation

[0054] In order to obtain the wave function of the particle, COMSOL uses the finite element method to solve the Schrödinger equation. This model adopts the steady-state research method, that is, the calculation of the model is independent of time. The Schrödinger equation in COMSOL is described as:

[0055]

[0056] in is the Hamiltonian corresponding to the total energy E of the Schrödinger equation, and ψ(r) is the variable wave function to be solved. In finite element calculations, in order to improve the accuracy and feasibility of the calculation results, boundary conditions need to be defined. In the one-dimensional model, we only defined open boundary conditions, while in the three-dimensional Josephson junction device, we not only defined open boundary conditions at both ends of the junction, but also defined periodic boundary conditions. The significance of setting open boundary conditions at both ends of the domain is that one end is used to limit the amplitude of the incident wave, and the other end is used to measure the outgoing wave. The equation of the open boundary condition in COMSOL is described as:

[0057]

[0058]

[0059]

[0060] Where ψ1 is the value of the solved wave function, calculated based on the defined open boundary conditions. ψ0 is defined as 1, indicating that the wave propagates in the defined direction with an amplitude of 1. n is the normal vector, and k0 represents the direction of the incident plane wave. In the one-dimensional square barrier model, the incident plane wave on the particle is specified to move at 1 rad / m along the x-axis. In the three-dimensional Josephson junction model, the particle is specified to move at 1 rad / m along the z-axis, the direction of electron transport.

[0061] These equations are solved based on finite element analysis to obtain the normalized wave function ψ(x) and the probability density |ψ(x)| 2 , we can call COMSOL built-in expressions to post-process the calculation results, where the built-in expressions schr.psi and schr.Pr_psi represent the wave function and probability density respectively. In the one-dimensional model, we extract the following Figure 4The images of the normalized wave function when particles with square barrier widths of 0.75nm, 1.25nm, 1.75nm, and 2.25nm tunnel through a one-dimensional square potential barrier shown in (ad).

[0062] We know that in the one-dimensional square barrier tunneling model, the transmission coefficient of the particle is equal to the square of the modulus of the wave function's outgoing amplitude divided by the incident amplitude, and the sum of the transmission coefficient and the reflection coefficient is 1. Therefore, in order to calculate the transmission coefficient of the one-dimensional square barrier tunneling model we simulated in COMSOL, we post-processed the wave function results and used a custom equation in COMSOL to calculate the particle's transmission coefficient.

[0063]

[0064] Where psi0 is the custom incident plane wave amplitude, with a value of 1, schr.open1.int(psi) represents the wave function amplitude calculated at the open boundary condition named open1 (the one-dimensional square barrier tunneling model constructed in this application can be viewed as a line segment divided into three regions, with open1 representing the rightmost endpoint of the line segment), and abs() represents the absolute value. We know the calculation principle of the transmission coefficient of the one-dimensional model, and by deriving and rewriting equation (4), we can customize the equation in COMSOL to calculate the tunneling probability ξ of a particle through the barrier layer in the three-dimensional model:

[0065] ξ=abs(aveop(schr.Pr_psi) / psi0)^2 (5)

[0066] Aveop() is an average value operator defined in the nonlocal coupling of physical fields. Its purpose is to calculate the average value of an expression in the selected geometric domain. Here, the geometric region we selected is the cross-sectional area of the top aluminum electrode. Therefore, aveop(schr.Pr_psi) calculates the average probability density in the cross-sectional area of the top aluminum electrode.

[0067] After calculating the tunneling probability of particles in the Josephson junction device, substituting this value into the Ambegaokar-Baratoff relationship, the critical current density J of the Josephson junction can be obtained. S and the constant temperature resistance R n :

[0068]

[0069] R n =2πh / (e 2 ξ×DOS) (7)

[0070] In the critical current density formula ξ is the tunneling probability, and DOS is the density of states at the Fermi level, which is defined as 1.62×10 14 cm -2 Δ(T) represents the superconducting energy gap in volts, T represents temperature, and the superconducting energy gap of metallic aluminum at 10mK is 2Δ=340μV. e is the charge, h is Planck's constant, and k B is the Boltzmann constant.

[0071] In COMSOL, we can customize equations (6) and (7) and define the parameters we need as global parameters in COMSOL to obtain the electrical properties related to the Josephson junction. Figure 5 and Figure 6 The tunneling probability and room temperature resistance results when the thickness of the insulating barrier layer changes from 0.5nm to 1.5nm are shown.

[0072] 3. Results Analysis

[0073] In the one-dimensional square barrier tunneling model, we compared the wave functions of particles with different barrier thicknesses. Figure 4 As shown in Figures (ad), the wave function amplitudes on the left and right sides of the barrier layer differ, demonstrating that even if a particle's energy is lower than that of the barrier region, it has a probability of passing through the barrier region and losing some energy. This means that the particle undergoes both transmission and reflection. This is the quantum tunneling effect, a phenomenon that cannot be explained by classical physics. As the thickness of the barrier layer changes, the output amplitude also changes. However, this change in amplitude affects the particle transmission coefficient, making the barrier layer thickness a key factor influencing particle tunneling.

[0074] Applying the above conclusions to the simulation of particle tunneling through a three-dimensional Josephson junction device, the thickness of the barrier layer determines the transmission coefficient, and the magnitude of the transmission coefficient reflects the strength of the electron transmission ability in the system. The larger the transmission coefficient, the higher the probability that the particle can tunnel through the barrier layer, and the stronger the conductivity of the system. Figure 5 This plot shows the tunneling probability as the barrier thickness changes. It shows that the tunneling probability does not decrease monotonically or linearly with increasing barrier thickness, but rather exhibits a fluctuating downward trend. This result is due to a very fine mesh. A coarser mesh would have resulted in more pronounced fluctuations in the tunneling probability. This conclusion is consistent with the tunneling probability results obtained by Kim's team using Josephson junction simulations.

[0075] Calculated by COMSOL Figure 6The curve of the Josephson junction's room-temperature resistance as the barrier layer thickness changes is shown. As the thickness of the aluminum oxide insulating film increases, the resistance value increases exponentially. This conclusion is consistent with the results of physical measurements of the Josephson junction's room-temperature resistance, indicating that the method of using COMSOL to simulate Josephson junction devices is effective.

[0076] In summary, the present invention adopts the COMSOL finite element analysis method to build a one-dimensional square barrier tunneling model and a three-dimensional Josephson junction device model. The barrier layer thickness of the two models is parametrically scanned to obtain the quantum tunneling effect of particles and the influence of the thickness of the aluminum oxide film on the electrical transport properties of the Josephson junction device. The present invention is based on the study of the influence of changing the barrier layer thickness on the tunneling probability, room temperature resistance and critical current density of the three-dimensional Josephson junction device by COMSOL. In building the three-dimensional Josephson junction model, we derived and imitated the formula for calculating the transmission coefficient and reflection coefficient of the one-dimensional square barrier tunneling model, customized the particle tunneling probability formula for calculating the three-dimensional model in COMSOL, and combined the formula with the Ambegaokar-Baratoff relationship to perform simulation calculations in COMSOL to obtain the critical current density and room temperature resistance of the three-dimensional Josephson junction device. Based on the above formula, a parametric sweep of the barrier layer thickness was performed, and it was found that thickness is an important factor affecting the electrical transport properties of the junction. The calculated result images were compared with simulation literature and physical experiment literature, and the results were found to be consistent, indicating that the study of the electrical transport properties of three-dimensional Josephson junction devices based on COMSOL is feasible and has guiding significance for actual physical experiments.

[0077] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A method for exploring the electrical transport properties of Al / α-Al2O3 / Al by changing the thickness of the barrier layer based on COMSOL, characterized in that: include: Step 1: Use the Schrödinger interface of the COMSOL Semiconductor Module to build a one-dimensional square barrier tunneling model and a three-dimensional Al / α-Al2O3 / Al Josephson junction device model. Step 2: Parametrically scan the barrier layer thickness of the two models to obtain the quantum tunneling effect of particles and the influence of the thickness of the aluminum oxide film on the electrical transport properties of the Josephson junction device; The step 2 includes: Step 2.1: Calculate the transmission coefficient of the one-dimensional square barrier tunneling model; Step 2.2: Calculate the tunneling probability of particles passing through barrier layers of different thicknesses in the 3D Al / α-Al2O3 / Al Josephson junction device model; Step 2.3: Substituting the tunneling probability into the Ambegaokar-Baratoff relation to obtain the critical current density and room temperature resistance of the Josephson junction device model; In step 2.1, the transmission coefficient of the one-dimensional square barrier tunneling model is calculated as follows: in Represents the transmission coefficient of the one-dimensional square barrier tunneling model, psi0 is the custom incident plane wave amplitude, schr.open1.int(psi) represents the wave function amplitude calculated at the open boundary condition named open1, open1 represents the rightmost endpoint of the one-dimensional square barrier tunneling model, and abs() represents taking the absolute value.

2. The method for exploring the electrical transport properties of Al / α-Al2O3 / Al by changing the barrier layer thickness based on COMSOL according to claim 1, characterized in that: In step 2.2, the tunneling probability of particles passing through the barrier layer in the three-dimensional Al / α-Al2O3 / Al Josephson junction device model is calculated as follows: ξ=abs(aveop(schr.Pr_psi) / psi0)^2 (5) Where aveop() is the average operator defined in the physical field nonlocal coupling, aveop(schr.Pr_psi) represents the average probability density at the cross-sectional area of the top aluminum electrode, abs() represents the absolute value, and psi0 is the custom incident plane wave amplitude.

3. The method for exploring the electrical transport properties of Al / α-Al2O3 / Al by changing the barrier layer thickness based on COMSOL according to claim 1, characterized in that: In step 2.3, the critical current density J of the Josephson junction device model is calculated as follows: S and the constant temperature resistance R n : R n =2πh / (e 2 (ξ×DOS) (7) in ξ is the tunneling probability, DOS is the density of states at the Fermi level, Δ(T) represents the superconducting energy gap in volts, T represents the temperature, e is the charge, h is the Planck constant, and k B is the Boltzmann constant.

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