A method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor

Abstract

The application discloses a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor modeling method, in which a function relationship among a ferroelectric layer electric field, a semiconductor surface charge area density and a ferroelectric body polarization intensity is obtained based on Maxwell equations and a potential displacement vector continuous equation of the ferroelectric-semiconductor layer, a numerical solution of a ferroelectric body dipole polarization differential equation is carried out through a second-order Rosenbrock formula, and a C-V curve of a thin film capacitor under a high-frequency state is simulated through voltage forward scanning and reverse scanning, so that the model can accurately depict the influence of capacitor parameters on the capacitance tunability. The application has great significance for parameter optimization design of a surface acoustic wave (SAW) filter based on the MFS tunable capacitor.

Description

Technical Field

[0001] This invention relates to the field of acoustic device manufacturing technology, and in particular to a method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor. Background Technology

[0002] With the development of technology, more and more mobile terminals are being invented. As mobile phone functions become increasingly rich, the demand for frequency bands is increasing year by year. However, each additional radio frequency (RF) band requires an additional RF front-end frequency selection filter in both the receiving and transmitting sections, inevitably increasing the overall power consumption of the phone. To increase the phone's frequency band without increasing the number of frequency selection filters, tunable RF filters are undoubtedly a good solution. Tunable RF filters are a type of filter that combines the excellent electrical performance of a single SAW filter with the ability to adjust the filter bandwidth or shift the passband. Compared to parallel-plate capacitors, MFS capacitors have higher power capacitance, smaller size, and shorter response time. Ferroelectric thin-film capacitors designed using the ferroelectric properties of ferroelectric materials have been widely used in tunable RF filters.

[0003] Current methods for fabricating thin-film capacitors are not only time-consuming, but also require extensive experience to determine the film thickness parameters. There is an urgent need for accurate MFS (Metal-Ferroelectric-Semiconductor-Semiconductor-Fluorescent) tunable capacitor models to describe their physical mechanisms and tunable capacitance principles. Optimizing parameters such as film thickness and dielectric constant, and fabricating samples based on these optimized parameters, can effectively improve experimental efficiency. However, most existing simulation models for ferroelectric thin-film capacitors neglect the influence of historical ferroelectric field strength on ferroelectric polarization, or have issues with the CV curves obtained during voltage reverse scanning, which affects the design of tunable capacitors. Therefore, this invention takes this as its starting point and, based on Maxwell's equations, invents a modeling method that can accurately describe tunable capacitors with metal-ferroelectric-semiconductor epitaxial heterostructures. Summary of the Invention

[0004] The purpose of this invention is to address the problems existing in the background technology by proposing a modeling method for tunable capacitors with metal-ferroelectric-semiconductor epitaxial heterostructures. In this modeling method, a CV model for thin-film capacitors under high-frequency conditions is derived based on Maxwell's equations, the Gaussian formula for semiconductor surfaces, and the second-order Rosenbrock formula. This model fully considers the influence of ferroelectric polarization intensity on ferroelectric capacitance, and corrects the calculation formula for ferroelectric capacitance in MFIS structures. Furthermore, based on the rigidity of the polarization differential equation and combined with the Rosenbrock formula, a numerical calculation method for the rigid differential equation is developed to achieve accurate solutions for the ferroelectric polarization intensity.

[0005] The technical solution of this invention provides a method for modeling a tunable capacitor with a metal-ferroelectric-semiconductor epitaxial heterostructure, comprising the following steps:

[0006] Step 1: The saturation polarization P of the ferroelectric material can be determined using the traditional Miller model. sat (E FE ) and ferroelectric polarization intensity P d (E FE The differential equations between ) are solved numerically using the second-order Rosenbrock formula;

[0007] Step 2: The electric field intensity E on the semiconductor surface can be derived from the Poisson equation for the electric field effect on the semiconductor surface. s With semiconductor surface potential Semiconductor carrier concentration N a Intrinsic carrier concentration n i The functional relationship between the two can be used to determine the surface charge density Q on the semiconductor surface according to Gauss's law. s With surface electric field intensity E s The functional relationship;

[0008] Step 3: The partial voltage E of the ferroelectric layer can be obtained from Maxwell's first equation and the continuity equation of the electric displacement vector of the ferroelectric-semiconductor layer. FE With semiconductor surface charge density Q s Ferroelectric polarization intensity P d (E FE The functional relationship between them;

[0009] Step 4: Following steps 2 and 3, the semiconductor surface potential can be obtained. Based on the physical properties of the semiconductor surface and MIS structure, the gate voltage Vg and the ferroelectric layer voltage E can be determined. FE and the surface potential of semiconductor depletion layer From the functional relationship, the voltage Vg can be obtained;

[0010] Step 5: Combine the semiconductor capacitance formulas for the accumulation and depletion regions under high-frequency operating conditions with the ferroelectric capacitance formula considering the polarization state, and finally obtain the equivalent total capacitance of the MFS capacitor based on the equivalent circuit.

[0011] The above modeling method can be implemented using MATLAB.

[0012] The algorithms for solving metal-ferroelectric-semiconductor (MFS) differential equations include, but are not limited to, the Rosenbrock algorithm, and numerical differentiation algorithms with accuracy greater than the second-order Rosenbrock algorithm are still applicable. The simulation model for this epitaxial heterostructure tunable capacitor is not limited to metal-ferroelectric-semiconductor MFS structures; it is also applicable to the simulation of capacitors with two MFS thin-film capacitors in series. Furthermore, the proposed method can be more broadly applied to the research of microwave phase shifters and resonators based on tunable capacitors.

[0013] Compared with the prior art, the present invention has the following beneficial technical effects:

[0014] This invention provides a modeling method for tunable capacitors with metal-ferroelectric-semiconductor epitaxial heterostructures. Compared with existing methods or technologies, this method offers the following advantages: It not only fully considers the influence of ferroelectric polarization intensity on ferroelectric capacitance and corrects the calculation formula for ferroelectric capacitance in MFIS structures, but also selects a suitable numerical calculation method for rigid differential equations based on the rigidity characteristics of the polarization differential equations, achieving accurate solutions for ferroelectric polarization intensity. This invention is not limited to metal-ferroelectric-semiconductor (MFS) structures; it is also applicable to the parameter optimization of capacitors with dual MFS thin-film capacitors in series. Furthermore, the proposed method can be more widely applied to the research of microwave phase shifters and resonators based on tunable capacitors. Attached Figure Description

[0015] Figure 1 This is a flowchart of the modeling method for the metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor of the present invention;

[0016] Figure 2 This illustrates the relationship between the polarization intensity of the ferroelectric material and the field strength of the ferroelectric layer in an embodiment of the present invention.

[0017] Figure 3 This illustrates the relationship between the semiconductor surface potential and the ferroelectric layer field strength in an embodiment of the present invention.

[0018] Figure 4 This illustrates the relationship between grid voltage and ferroelectric layer field strength in an embodiment of the present invention.

[0019] Figure 5 This illustrates the relationship between the equivalent capacitance of the ferroelectric material and the grid voltage in an embodiment of the present invention.

[0020] Figure 6 This illustrates the relationship between the semiconductor equivalent capacitance and the grid voltage in an embodiment of the present invention.

[0021] Figure 7 This illustrates the relationship between the MFS equivalent capacitance and the grid voltage in an embodiment of the present invention.

[0022] Figure 8 for Figure 7The result after normalization. Detailed Implementation

[0023] Example 1

[0024] First, assign a value to the variable: d FE =140nm,ε FE =3100,E c =106 kV / cm, P s =3.6μc / cm 2 ,P r =3.11μc / cm 2 ,ε s =30,ε o =8.85×10 -12 F / m,N a =10 17 cm -3 ,n i =10 10 cm -3 T = 300K, q = 1.6 × 10 -19 c,k = 1.38 × 10 -23 .

[0025] Based on the above, β = 38.6473v is obtained. -1 δ=40.5052kv / cm,L B =2.07×10 -11 m. Reference Figure 1 The modeling process is as follows:

[0026] Step 1: The saturation polarization P of the ferroelectric material can be determined using the Miller model. sat (E FE ) and ferroelectric polarization intensity P d (E FE The differential equations between the two sides are solved numerically using the second-order Rosenbrock formula, and the boundary conditions are determined by the endpoints of the saturation polarization curve.

[0027] Step 2: The electric field intensity E on the semiconductor surface can be derived from the Poisson equation for the electric field effect on the semiconductor surface. s With semiconductor surface potential Semiconductor carrier concentration N a Intrinsic carrier concentration n i The functional relationship between the two can be used to determine the surface charge density Q on the semiconductor surface according to Gauss's law. s With surface electric field intensity E s The functional relationship;

[0028] Step 3: The partial voltage E of the ferroelectric layer can be obtained from Maxwell's first equation and the continuity equation of the electric displacement vector of the ferroelectric-semiconductor layer.FE With semiconductor surface charge density Q s Ferroelectric polarization intensity P d (E FE The functional relationship between them;

[0029] Step 4: Following steps 2 and 3, the semiconductor surface potential can be obtained. Based on the physical properties of the semiconductor surface and MIS structure, the gate voltage Vg and the ferroelectric layer voltage E can be determined. FE and the surface potential of semiconductor depletion layer From the functional relationship, the voltage Vg can be obtained;

[0030] Step 5: Combine the semiconductor capacitance formulas for the accumulation and depletion regions under high-frequency operating conditions with the ferroelectric capacitance formula considering the polarization state, and finally obtain the equivalent total capacitance of the MFS capacitor based on the equivalent circuit.

[0031] In step 1, the saturation polarization intensity is:

[0032]

[0033] Among them, P sat→ (E FE The corresponding saturation polarization intensity P during positive voltage scanning sat← (E FE The corresponding saturation polarization intensity of the voltage reverse scan; P s E represents the spontaneous polarization intensity. FE E represents the electric field strength of a ferroelectric material. c For coercive field strength, δ is related to E c P s P r The relevant parameter, δ=E c / ln((1+P r / P s ) / (1-P r / P s For the unsaturated polarization state of the ferroelectric layer, the electric dipole polarization affects the electric field E. FE The differential is:

[0034]

[0035] Where Γ is 1-tanh(((P) d -P sat ) / (ξP s -P d )) 2 ),when When the voltage is in positive sweep, ξ = 1; when When the voltage is reversed, ξ = -1.

[0036] The solution of differential equations by the Rosenbrock method is based on the Runge - Kutta method. When solving stiff differential equations, the order of magnitude of the number of multiplications required for each iteration of the implicit Runge - Kutta method is s 3 m 3 , where s is the order of the implicit Runge - Kutta equation and m is the number of equations, resulting in a huge workload. The Rosenbrock method can greatly reduce the workload of the implicit Runge - Kutta algorithm. For convenient numerical solution, formula (2) is appropriately transformed as follows:

[0037]

[0038] The specific solution form of the Rosenbrock method is:

[0039]

[0040]

[0041] where h represents the calculation step size, the node \(t_n = nh\), I represents the appropriate - dimensional identity matrix, j represents the Jacobian matrix, \(f t (t n ,u n ) the u n represents the numerical approximation of \(u(t n )\), and \(k i represents the numerical approximation of the extreme value;

[0042] By联立公式(1)(2)(3)建立铁电体极化强度微分方程,使用罗森布罗克常微分方程数值求解方法,可精确求解铁电极化强度随铁电层电场变化的数值关系,记为:

[0043] P d (E FE )=f1(E FE ) (4)

[0044] 求解的铁电体极化强度P d (E FE )如 Figure 2 所示,与MFIS电容器表现出铁电极化的不饱和磁滞回线相反,基于半导体氧化物的MFS电容器表现出饱和磁滞回线( Figure 2 )。

[0045] In step 2, the electric field strength on the semiconductor surface is:

[0046]

[0047] It should be noted that there seems to be some incorrect or incomplete expressions in the original text, such as "联立公式(1)(2)(3)建立铁电体极化强度微分方程" which is not properly translated in the current translation due to the unclear expression in Chinese. You may need to check and correct the original text for a more accurate translation.Where β = q / kT, q is the unit charge, k is the Boltzmann constant, and T is the thermodynamic temperature; Debye length φ s n is the surface potential of the semiconductor. i This refers to the intrinsic carrier concentration.

[0048] According to Gauss's law, the surface charge density of a semiconductor is:

[0049]

[0050] Use sgn(φ) s This indicates that the electric field strength is positive only when it points into the semiconductor, otherwise it is negative.

[0051] In step 3, the continuity equations of the electric displacement vectors of each layer are obtained from Maxwell's first equation:

[0052] D = ε o ε FE E FE +P d (E FE )=ε o ε s E s (7)

[0053] From formula (5), the electric field strength of the ferroelectric material can be obtained as follows:

[0054] E FE =-(Q s (φ s )+P d (E FE )) / ε o ε FE (8)

[0055] By combining formulas (4), (6), and (8), the numerical relationship between the semiconductor surface potential and the ferroelectric layer electric field can be solved, denoted as:

[0056] φ s (E FE )=f2(E FE (9)

[0057] Solving for the semiconductor surface potential φ s (E FE )like Figure 3 As shown in the figure, the presence of ferroelectric polarization leads to φ s (E FE The lag of the curve.

[0058] The gate voltage in step 4 is:

[0059] V g =V f +φ s (10)

[0060] Among them, V f The electric field of a ferroelectric body can be approximated as E. FE dE FE By combining formulas (9) and (10), the numerical relationship between the gate voltage and the electric field of the ferroelectric layer can be obtained, denoted as:

[0061] V g (E FE )=f3(E FE (11)

[0062] Solving for the grid voltage V g (E FE )like Figure 4 As shown, the fact that the forward and reverse scan curves do not overlap proves the existence of polarization hysteresis.

[0063] In step 5, the semiconductor capacitance formulas for the accumulation and depletion regions under high-frequency operating conditions are as follows:

[0064]

[0065] The formula for ferroelectric capacitance is:

[0066]

[0067] Therefore, the total capacitance value is:

[0068] C total =(C s -1 +C FE -1 ) -1 (14)

[0069] Combining formulas (4), (9), (12), and (13), we can obtain the relationship between the semiconductor capacitance and the ferroelectric capacitance and the electric field of the ferroelectric material, denoted as C. s (E FE ),C FE (E FE )like Figure 5 As shown in Figure 6; E can be obtained from formula (11). FE =f3 -1 (V g Finally, the relationship between capacitance and grid voltage C is obtained. total (V gThe shape of the CV curve of an MFS capacitor is related to the direction of the voltage scan, because the direction of the voltage scan affects the charge distribution on the semiconductor surface and the charge trapping in the oxide layer. At high frequencies, the CV curve of an MFS capacitor only has a hole accumulation region and an inversion region, without a depletion region. When the voltage scans from -6V to 6V, the semiconductor surface first experiences hole accumulation, then gradually inversion. The capacitance is initially determined by the oxide layer capacitance, and then by the series connection of the oxide layer capacitance and the depletion layer capacitance. Therefore, the capacitance decreases as the voltage increases. When the voltage scans from 6V to -6V, the semiconductor surface first experiences inversion, then gradually hole accumulation. The capacitance is initially determined by the series connection of the oxide layer capacitance and the depletion layer capacitance, and then by the oxide layer capacitance. Therefore, the capacitance decreases as the voltage decreases. Figure 7 As shown in Figure 8.

[0070] The proposed algorithms for solving metal-ferroelectric-semiconductor (MFS) differential equations include, but are not limited to, the Rosenbrock algorithm. Numerical differentiation algorithms with accuracy greater than that of the second-order Rosenbrock algorithm are also applicable.

[0071] This simulation model for an epitaxial heterostructure tunable capacitor is not limited to metal-ferroelectric-semiconductor (MFS) structures; it is also applicable to the simulation of capacitors with two MFS thin-film capacitors connected in series. Furthermore, the proposed method can be more widely applied to the research of microwave phase shifters and resonators based on tunable capacitors.

[0072] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited thereto. Various changes can be made within the scope of knowledge possessed by those skilled in the art without departing from the spirit of the present invention.

Claims

1. A method for modeling a tunable capacitor with a metal-ferroelectric-semiconductor epitaxial heterostructure, characterized in that, The specific steps include the following: S1. Determining ferroelectric saturation polarization using the Miller model. With ferroelectric polarization intensity The differential equations between the two sides are solved numerically using the second-order Rosenbrock formula, and the boundary conditions are determined by the endpoints of the saturation polarization curve. S2. Derive the electric field strength on the semiconductor surface using the Poisson equation for the semiconductor surface electric field effect. With semiconductor surface potential Semiconductor carrier concentration Intrinsic carrier concentration The functional relationship; determining the surface charge density of the semiconductor surface based on Gauss's law. With surface electric field strength The functional relationship; S3. Obtain the voltage divider of the ferroelectric layer using Maxwell's first equation and the continuity equation of the electric displacement vector of the ferroelectric-semiconductor layer. With semiconductor surface charge density Ferroelectric polarization intensity Functional relationship between them; S4. The semiconductor surface potential is obtained through S2 and S3. Based on the physical properties of the semiconductor surface and MIS structure, it can be known that the gate voltage Vg is related to the voltage divider of the ferroelectric layer. and the surface potential of semiconductor depletion layer The functional relationship is then used to obtain the voltage Vg; S5. By combining the semiconductor capacitance formulas for the accumulation and depletion regions under high-frequency operating conditions and the ferroelectric capacitance formula considering the polarization state, the equivalent total capacitance of the MFS capacitor can be finally obtained based on the equivalent circuit.

2. The method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor according to claim 1, characterized in that, In S1, the saturation polarization intensity is: (1) in, The corresponding saturation polarization intensity for positive voltage scanning. The saturation polarization intensity corresponding to the voltage reverse scan; For spontaneous polarization intensity, The electric field strength of a ferroelectric material. To coercive field strength, To and , , Relevant parameters, For the unsaturated polarization state of the ferroelectric layer, the electric dipole polarization affects the electric field. The differential is: (2) in, for ,when At that time, i.e., voltage forward sweep, ;when At that time, i.e., voltage reverse scan, .

3. The method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor according to claim 2, characterized in that, In S1, the Rosenbrock method is used to transform formula (2) into: （3） The Rosenbrock method has the following specific solution form: ； ； in, Indicates the calculation step size, node , Represent an identity matrix of appropriate dimension. Represents the Jacobian matrix, middle Indicates to The numerical approximation is... This indicates a numerical approximation of the level value; By combining formulas (1), (2), and (3), a differential equation for the polarization intensity of the ferroelectric material is established. Using the Rosenbrock ordinary differential equation numerical solution method, the numerical relationship between the polarization intensity of the ferroelectric material and the electric field of the ferroelectric layer can be accurately solved, denoted as: （4）。 4. The method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor according to claim 3, characterized in that, In S2, the electric field intensity on the semiconductor surface is: （5） in, , For a unit charge, Boltzmann's constant, Thermodynamic temperature; Debye length , The potential of the semiconductor surface. This refers to the intrinsic carrier concentration. ; According to Gauss's law, the surface charge density of a semiconductor is: （6） use This indicates that the electric field strength is positive only when it points into the semiconductor, and negative otherwise.

5. The method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor according to claim 4, characterized in that, In S3, the continuity equations of the electric displacement vectors of each layer are obtained from Maxwell's first equation: （7） From formula (5), the electric field strength of the ferroelectric material can be obtained as follows: （8） By combining formulas (4), (6), and (8), the numerical relationship between the semiconductor surface potential and the ferroelectric layer electric field can be solved, denoted as: （9）。 6. The method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor according to claim 5, characterized in that, The gate voltage in S4 is: （10） in, The electric field of a ferroelectric body can be approximated as: By combining formulas (9) and (10), the numerical relationship between the gate voltage and the electric field of the ferroelectric layer can be obtained, denoted as: （11）。 7. The method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor according to claim 6, characterized in that, In S5, the semiconductor capacitance formulas for the accumulation and depletion regions under high-frequency operating conditions are: （12） The formula for ferroelectric capacitance is: （13） Therefore, the total capacitance value is: （14） Combining formulas (4), (9), (12), and (13), we can obtain the relationship between the semiconductor capacitance and the ferroelectric capacitance and the electric field of the ferroelectric material, denoted as From formula (11), we can obtain Finally, the relationship between capacitance and grid voltage was obtained. .

8. The method for modeling a metal-ferroelectric-semiconductor epitaxial heterostructure tunable capacitor according to claim 1, characterized in that, Algorithms for solving metal-ferroelectric-semiconductor differential equations include the Rosenbrock algorithm.

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