CN102542077A - Parameter extraction method of AlGaN/GaN HEMT small-signal model - Google Patents

Parameter extraction method of AlGaN/GaN HEMT small-signal model Download PDF

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CN102542077A
CN102542077A CN2010105890285A CN201010589028A CN102542077A CN 102542077 A CN102542077 A CN 102542077A CN 2010105890285 A CN2010105890285 A CN 2010105890285A CN 201010589028 A CN201010589028 A CN 201010589028A CN 102542077 A CN102542077 A CN 102542077A
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CN102542077B (en
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刘新宇
蒲颜
庞磊
袁婷婷
罗卫军
陈晓娟
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Abstract

The invention relates to a parameter extraction method of an algan/gan hemt small signal model, belonging to the technical field of integrated circuits. The parameter extraction method is an improvement on the basis of the traditional parameter extraction method, and adopts an open-circuit de-embedding pattern to extract peripheral parasitic parameters and introduces a gate-end Schottky resistorExtracting parasitic resistance and inductance, and introducing drain terminal delay factorExtracting internal intrinsic parameters to ensure that the extracted parameters are all positive values and all have physical significance, thereby improving s in s parameters of small signal parameters11And s22In the course of parameter extractionAndboth terms are often prone to negative values,andthese improvements result in a substantial improvement in the accuracy of the extracted parameters, substantially eliminating their possibility of negative values.

Description

Parameter extraction method of AlGaN/GaN HEMT small-signal model
Technical Field
The invention relates to a parameter extraction method, in particular to a parameter extraction method of an AlGaN/GaN HEMT small-signal model, and belongs to the technical field of integrated circuits.
Background
The AlGaN/GaN HEMT small signal model is the basis for building a large signal model, and the extraction method of the small signal model parameters is the main factor for determining whether the small signal model is accurate. The small-signal model parameters are determined by the small-signal model topology, including peripheral parasitic parameters and internal intrinsic parameters. In general, the peripheral parasitic parameters are linear elements, i.e. not changing with the external bias voltage and frequency, and mainly include the series parasitic resistance Rg,Rd,Rs(ii) a Series parasitic inductance Lg,Ld,LsPeripheral parasitic capacitance Cpg,CpdAnd Cpgd. The intrinsic parameter of the device is a nonlinear element which changes with the change of external bias voltage and frequency and mainly comprises an internal gate capacitor Cgs、CgdSource-drain current source IdsFrom transconductance gmAnd its delay factor taumCharacterization, source-drain conductance gdsAnd its delay factor tauds. Drain-source capacitance CdsAnd gate source channel resistance RiThe variation with the external bias voltage is small and can be regarded as a linear parameter. In the case of small signal excitation, the variation of the internal intrinsic element can be equivalent to a linearly varying element, so that all parameters of the small signal model have determined values under a fixed bias state to characterize the high-frequency characteristics of the device under a specific state.
The small signal model is related to the topological structure of the device, the topological structure of the device represents the physical characteristics of the device, namely, each small signal model parameter has specific physical significance, so that the physical structure characteristics and specific physical explanations of the device can be reflected, the physical structure characteristics and the specific physical explanations are closely related to each step of process parameters in the manufacturing process of the device, the change of the physical parameters can cause the change of the small signal model parameters, and the small signal model parameters can also guide the process steps, improve the structure of the device and guide the improvement direction of the performance of the device; the small signal model reflects the high-frequency characteristic under the specific bias and is a necessary step for establishing the large signal model, so that the extraction of the small signal model parameters also relates to the accuracy of the large signal model, and the large signal model can reflect the small signal characteristic under the specific bias.
For the GaN HEMT device, because the device is a new material and a new device, the process accuracy of each step is important to monitor in places different from the traditional process steps, and after the device is manufactured, the equivalent circuit model can integrally represent the characteristics of the device, reflect the performance of the device and integrally evaluate various characteristics of the device in the process flow. The small signal equivalent circuit can simulate the S parameter of a specific bias state in the circuit to obtain the gains of the device and the circuit, and can be used for designing small signal amplifiers and the like. Therefore, the method is significant for the research of the small-signal equivalent circuit, and therefore the method for extracting the parameters of the small-signal equivalent circuit becomes important.
Disclosure of Invention
The invention aims to link the physical topology of a device with equivalent circuit parameters and provide a necessary foundation for establishing a large signal model of the device, and provides a parameter extraction method of an AlGaN/GaN HEMT small signal model.
The technical scheme for solving the technical problems is as follows: a parameter extraction method for an AlGaN/GaN HEMT small signal model comprises the following steps:
step 10: measuring scattering parameter S of peripheral open-circuit de-embedding circuit, converting to obtain admittance parameter Y, and calculating peripheral parasitic capacitance Cpg、CpdAnd CpgdThe peripheral open circuit de-embedding circuit comprises a peripheral parasitic capacitance Cpg、CpdAnd CpgdSaid peripheral parasitic capacitance CpgdSeries CpgAnd CpdTo (c) to (d);
step 20: at Vgs>0,VdsUnder the bias state of 0V, the gate voltage values of two groups of AlGaN/GaN HEMT devices are respectively selected to be Vgs1And Vgs2Testing to obtain two groups of current values I which respectively correspond to the two groups of gate voltage values and are less than 10mAgs1And Igs2Value of current Igs1And Igs2Are relatively close, and then the gate voltage values are measured to be V respectivelygs1And Vgs2The S parameter is converted to obtain an impedance parameter Z, and then the values of the series parasitic resistance and the series parasitic inductance are calculated;
step 30: measuring S parameter of AlGaN/GaN HEMT device in bias state, and removing peripheral parasitic capacitance C calculated in step 10pg、CpdAnd CpgdAnd the series parasitic resistance R obtained in step 20g、Rd、RsAnd series parasitic inductance Lg、Ld、LsObtaining intrinsic S parameters of the internal parameters, converting the intrinsic S parameters to obtain Y parameters, and calculating to obtain internal intrinsic parameter gate capacitance C in a bias stategs、CgdTransconductance gmAnd its delay factor taumSource drain conductance gdsAnd its drain terminal delay factor taudsDrain-source capacitance CdsAnd gate source channel resistance RiThe numerical value of (c).
On the basis of the technical scheme, the invention can be further improved as follows.
Further, the step 10 calculates the peripheral parasitic capacitance Cpg、CpdAnd CpgdThe process of numerical values of (1) includes: converting the S parameter to obtain a Y parameter according to formulas (1) - (3)
Im(Y11)=ω(Cpg+Cpgd) (1)
Im(Y12)=Im(Y21)=-ωCpgd (2)
Im(Y22)=ω(Cpd+Cpgd) (3)
Calculating to obtain the peripheral parasitic capacitance Cpg、CpdAnd CpgdRespectively are
<math><mrow> <msub> <mi>C</mi> <mi>pgd</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>,</mo> <msub> <mi>C</mi> <mi>pg</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>-</mo> <msub> <mi>C</mi> <mi>pgd</mi> </msub> <mo>,</mo> <msub> <mi>C</mi> <mi>pd</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>-</mo> <msub> <mi>C</mi> <mi>pgd</mi> </msub> <mo>.</mo> </mrow></math>
Further, the gate voltage values of the two groups of AlGaN/GaN HEMTs arbitrarily selected in the step 20 are all greater than the schottky voltage of the AlGaN/GaN HEMTs, and the current values corresponding to the gate voltage values of the two groups of AlGaN/GaN HEMTs are all less than 10 mA.
Further, said step 20 is at Vgs>0,VdsUnder the bias state of 0V, the equivalent circuit of the AlGaN/GaN HEMT device comprises a series parasitic resistor Rg、RdAnd RsSeries parasitic inductance Lg、LdAnd LsMultiple gate distributed resistance Δ RgsMultiple gate terminal distributed capacitance Δ CgMultiple channel distributed resistance Δ RchAnd a plurality of channel distributed capacitances Δ CdsWherein the gate terminal has a distributed resistance Δ RgsAnd gate terminal distributed capacitance Δ CgAre connected in parallel to form a grid end distributed parallel unit, and a plurality of grid end distributed parallel units are connected in parallel with a series parasitic resistor RgAre connected in series, the series parasitic resistance RgAnd series parasitic inductance LgAre connected in series, and the channel distribution resistance is delta RchAnd channel distributed capacitance Δ CdsAre connected in parallel to form a channel distribution parallel unit, and one end of the plurality of channel distribution parallel units is connected with a series parasitic resistor R after being connected in seriesdAre connected in series, and the other end is connected with a series parasitic resistor RsAre connected in series, the series parasitic resistance RdAnd series parasitic inductance LdAre connected in series, the series parasitic resistance RsAnd series parasitic inductance LsAre connected in series.
Further, said step 20 is to Vgs1And Vgs2Two groups of Z parameters obtained by the transformation of the two groups of S parameters are respectivelyAnd
Figure BDA0000038361470000042
according to the formula
n = Re ( Z 11 1 ) - Re ( Z 11 2 ) kT / ( qI gs 1 - q I gs 2 ) And R gs = nkT qI gs
n and R are obtained by calculationgs,RgsThe schottky resistance at the gate end can be obtained according to the transmission line equivalent equation as follows:
<math><mrow> <msub> <mi>Z</mi> <mn>11</mn> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>g</mi> </msub> <mo>+</mo> <mfrac> <msub> <mi>Z</mi> <mn>0</mn> </msub> <mrow> <mi>tanh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>j&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>L</mi> <mi>g</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow></math>
<math><mrow> <msub> <mi>Z</mi> <mn>12</mn> </msub> <mo>=</mo> <msub> <mi>Z</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>Z</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>cosh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sinh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>j&omega;</mi> <msub> <mi>L</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow></math>
<math><mrow> <msub> <mi>Z</mi> <mn>22</mn> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>d</mi> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mrow> <mn>2</mn> <mi>Z</mi> </mrow> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>cosh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sinh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>j&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>L</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow></math>
wherein Δ Rch=Rch×Δx,ΔCds=Cds×Δx,ΔRgs=Rgs×Δx,ΔCg=CgX Δ x, Δ x is an infinitesimal length along the gate length direction.
Wherein: z0Is the characteristic impedance, γ L is the product of the transmission constant and the length along the gate length,
<math><mrow> <msub> <mi>Z</mi> <mn>0</mn> </msub> <mo>=</mo> <msqrt> <mfrac> <mrow> <msub> <mi>R</mi> <mi>ch</mi> </msub> <mo>/</mo> <mo>/</mo> <msub> <mi>C</mi> <mi>ds</mi> </msub> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <msub> <mi>R</mi> <mi>gs</mi> </msub> <mo>+</mo> <mi>j&omega;</mi> <msub> <mi>C</mi> <mi>g</mi> </msub> </mrow> </mfrac> </msqrt> </mrow></math>
<math><mrow> <mi>&gamma;L</mi> <mo>=</mo> <msqrt> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>ch</mi> </msub> <mo>/</mo> <mo>/</mo> <msub> <mi>C</mi> <mi>ds</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>/</mo> <msub> <mi>R</mi> <mi>gs</mi> </msub> <mo>+</mo> <mi>j&omega;</mi> <msub> <mi>C</mi> <mi>g</mi> </msub> <mo>)</mo> </mrow> </msqrt> </mrow></math>
since (gamma L)2|<<1,ωRchCds<<1,ωRgsCg1, the Z parameter forms of the formulas (7) to (9) can be obtained through simplification,
Z11=Rs+Rg+RgsgRch+jω(Ls+Lg) (7)
Z12=Z21=Rs+αRch+jωLs (8)
Z22=Rs+Rd+2αRch+jω(Ls+Ld) (9)
whereinDue to Re (Z)11) And
R gs = nkT q I gs
proportional relation, so obtained by measurement
Figure BDA0000038361470000051
The intercept of the linear fit is Rs+RggRchAnd at the same time due toWherein VthIs measured as a threshold voltage
Figure BDA0000038361470000053
The fitting intercept of the low leakage pressure is Rd+RsThus, in combination with the above formulas (7) to (9), the corresponding resistance value can be calculated, and then the series parasitic resistance and the series parasitic inductance are obtained by calculating the real part and the imaginary part respectively:
Rg=real(Z11)-real(Z12)+Rch/6,Rd=real(Z22)-real(Z12)-Rch/2,
Rs=real(Z12)-Rch/2,Lg=imag(Z11-Z12)/ω,
Ld=imag(Z22-Z12)/ω,Ls=imag(Z12)/ω,
wherein alpha isgAnd alpha is typically 1/3 and 1/2.
Further, the step 30 calculates to obtain the internal intrinsic parameter value of the gate capacitance C under a specific bias stategs、CgdTransconductance gmAnd its delay factor taumSource drain conductance gdsAnd its drain terminal delay factor taudsDrain-source capacitance CdsAnd gate source channel resistance RiThe process comprises the following steps: the intrinsic S parameter of the internal parameter is converted to obtain the Y parameter according to the formulas (10) to (13)
<math><mrow> <msub> <mi>Y</mi> <mn>11</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>i</mi> </msub> <msubsup> <mi>C</mi> <mi>gs</mi> <mn>2</mn> </msubsup> <msup> <mi>&omega;</mi> <mn>2</mn> </msup> </mrow> <mi>D</mi> </mfrac> <mo>+</mo> <mi>j&omega;</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>C</mi> <mi>gs</mi> </msub> <mi>D</mi> </mfrac> <mo>+</mo> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow></math>
Y12=-jωCgd (11)
<math><mrow> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>g</mi> <mi>m</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>j&omega;</mi> <msub> <mi>R</mi> <mi>i</mi> </msub> <msub> <mi>C</mi> <mi>gs</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow></math>
<math><mrow> <msub> <mi>Y</mi> <mn>22</mn> </msub> <mo>=</mo> <msub> <mi>g</mi> <mi>ds</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>&tau;</mi> <mi>ds</mi> </msub> </mrow> </msup> <mo>+</mo> <mi>j&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>ds</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow></math>
Wherein, <math><mrow> <mi>D</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>&omega;</mi> <mn>2</mn> </msup> <msubsup> <mi>C</mi> <mi>gs</mi> <mn>2</mn> </msubsup> <msubsup> <mi>R</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>,</mo> </mrow></math>
and calculating the values of the intrinsic parameters in the parts respectively as
gds=real(Y22), <math><mrow> <msub> <mi>&tau;</mi> <mi>ds</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&omega;</mi> </mfrac> <msqrt> <mrow> <mo>(</mo> <mn>2</mn> <mo>-</mo> <mn>2</mn> <mo>*</mo> <mi>real</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>g</mi> <mi>ds</mi> </msub> <mo>)</mo> </mrow> </msqrt> <mo>,</mo> </mrow></math>
<math><mrow> <msub> <mi>C</mi> <mi>gs</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mi>&omega;</mi> </mfrac> <mo>*</mo> <mrow> <mo>(</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mi>D</mi> <mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>*</mo> <mrow> <mo>(</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow></math>
<math><mrow> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>,</mo> <msub> <mi>C</mi> <mi>ds</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>+</mo> <msub> <mi>g</mi> <mi>ds</mi> </msub> <mo>*</mo> <msub> <mi>&tau;</mi> <mi>ds</mi> </msub> <mo>,</mo> </mrow></math>
g m = ( 1 + D 2 ) * ( real ( Y 21 ) 2 + ( imag ( Y 21 ) - imag ( Y 12 ) ) 2 ) ,
<math><mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mi>&omega;</mi> </mfrac> <mo>*</mo> <mi>arctan</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>D</mi> <mo>*</mo> <mi>real</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>real</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>D</mi> <mo>*</mo> <mrow> <mo>(</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>.</mo> </mrow></math>
The invention has the beneficial effects that: the parameter extraction method of the AlGaN/GaN HEMT small-signal model can obtain specific parameters corresponding to the topological structure of the device, so that the physical mechanism of the device and the parameters in specific process steps are linked together, the physical process is conveniently analyzed, the problems in the device manufacturing process are conveniently found, different results can be compared, and the method has reference value for improving the process steps and updating the device structure; after small signal parameters are extracted, a small signal equivalent model of the device can be established for circuit design under small signal application, gain and S parameters under a specific bias state can be obtained through simulation, and meanwhile, a foundation is laid for establishment of a large signal model, so that the accurate extraction method of the small signal parameters has a great research value.
Drawings
Fig. 1 is a schematic structural view of an AlGaN/GaN HEMT device used in the parameter extraction method for the AlGaN/GaN HEMT small-signal model according to the embodiment of the present invention;
FIG. 2 is a parametric equivalent circuit diagram of FIG. 1;
FIG. 3 is an equivalent circuit diagram of an open de-embedding pattern for extracting peripheral parasitic capacitance parameters in the parameter extraction method for the AlGaN/GaN HEMT small-signal model according to the embodiment of the invention;
FIG. 4 is an equivalent circuit diagram of device channel distribution parameters in the parameter extraction method for the AlGaN/GaN HEMT small-signal model according to the embodiment of the invention;
FIG. 5 is a high-frequency characteristic diagram of a device with parameters extracted in the parameter extraction method for the AlGaN/GaN HEMT small-signal model according to the embodiment of the invention;
FIG. 6 shows a gate Schottky resistor R introduced into the parameter extraction method of the AlGaN/GaN HEMT small-signal model according to the embodiment of the inventiongsThen, the extracted small signal equivalent circuit parameter pair S11The improvement of (2) is compared with a schematic diagram;
FIG. 7 shows a delay factor τ introduced into the source-drain conductance in the parameter extraction method of the AlGaN/GaN HEMT small-signal model according to the embodiment of the present inventiondsThen, the extracted small signal equivalent circuit parameter pair S22The improvement of (2) is compared with the schematic diagram.
Detailed Description
The principles and features of this invention are described below in conjunction with the following drawings, which are set forth by way of illustration only and are not intended to limit the scope of the invention.
Fig. 1 is a schematic structural diagram of an AlGaN/GaN HEMT device used in a parameter extraction method for an AlGaN/GaN HEMT small-signal model according to an embodiment of the present invention, and a specific example to be described later is a method for extracting parameters by using the device structure. The extraction of small signal parameters is an essential link in the modeling of the FET semiconductor device, the accurate extraction of peripheral parasitic parameters can influence the accuracy of internal nonlinear parameters, the extraction of small signal models is the basis for the establishment of large signal models, and the large signal models can be established more accurately only by ensuring that all small signal parameter values are accurate and reasonable.
Fig. 2 is an equivalent circuit diagram of fig. 1. As shown in FIG. 2, the equivalent circuit diagram includes a peripheral parasitic capacitor Cpg,Cpd,CpgdSeries parasitic resistance Rg,Rd,RsSeries parasitic inductance Lg,Ld,LsGrid capacitance Cgs、CgdTransconductance gmAnd its delay factor taumSource drain conductance gdsAnd its drain terminal delay factor taudsDrain-source capacitance CdsAnd gate source channel resistance RiVi is the capacitance CgsThe voltage across, Vd, being the capacitance CdsThe voltage across the two terminals, and the method for extracting the parameters herein, are also illustrative of how the above parameters are extracted in this equivalent circuit diagram.
FIG. 3 is an equivalent circuit diagram of an open de-embedding pattern in the parameter extraction method of the AlGaN/GaN HEMT small-signal model according to the embodiment of the invention, and the equivalent circuit diagram has three peripheral parasitic capacitance parameters Cpg,Cpd,Cpgd
FIG. 4 is an equivalent circuit diagram of device channel distribution parameters in the parameter extraction method for the AlGaN/GaN HEMT small-signal model according to the embodiment of the invention, and as shown in FIG. 4, the gate distributed resistance and the distributed capacitance are respectively Δ RgsAnd Δ CgThe distributed resistance and the distributed capacitance of the channel are respectively delta RchAnd Δ CdsThe solution equation of the equivalent circuit can be obtained by simplifying the equivalent equation of the transmission line, and then the numerical values of the series parasitic resistance and the series parasitic inductance are obtained by calculation.
Fig. 5 is a high-frequency characteristic diagram of a device with parameters extracted in the parameter extraction method for the AlGaN/GaN HEMT small-signal model according to the embodiment of the present invention. In order to verify the accuracy of the extracted small signal parameters under the fixed bias state, the effect of comparing the measured S parameters with the model fitting S parameters is required, and the formula reflecting the physical significance of the device is used
<math><mrow> <msub> <mi>f</mi> <mi>T</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>g</mi> <mi>m</mi> </msub> <mrow> <mn>2</mn> <mi>&pi;</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>gs</mi> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>)</mo> </mrow> <mo>[</mo> <mn>1</mn> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>g</mi> <mi>ds</mi> </msub> <mo>]</mo> <mo>+</mo> <msub> <mi>C</mi> <mi>gd</mi> </msub> <msub> <mi>g</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow></math>
And
<math><mrow> <msub> <mi>f</mi> <mi>max</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>f</mi> <mi>T</mi> </msub> <mrow> <mn>2</mn> <msqrt> <mrow> <mo>(</mo> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>g</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>g</mi> <mi>ds</mi> </msub> <mo>+</mo> <mn>2</mn> <mi>&pi;</mi> <msub> <mi>f</mi> <mi>T</mi> </msub> <msub> <mi>C</mi> <mi>gd</mi> </msub> <msub> <mi>R</mi> <mi>g</mi> </msub> <mo>)</mo> </mrow> </msqrt> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow></math>
the prediction of the high frequency characteristics of the device for the extracted parameters, compared to the measured high frequency characteristics, can be used as a criterion for determining whether the extracted parameters are accurate, and it can be seen from fig. 5 that the measured cut-off frequency f of the devicetAt 36GHz and a maximum oscillation frequency fmaxAt 58GHz, using the parameters extracted in Table 1 below, f predicted by equations (14) and (15)tIs 34.9GHz, fmaxThe frequency is 56.4GHz, so that the small-signal equivalent circuit parameters extracted by the method have physical significance, the fitting S parameter effect is good, the high-frequency characteristics of the device can be well predicted, and the extracted parameters are very accurate.
FIG. 6 shows a gate schottky resistor R is introduced into the parameter extraction method of the AlGaN/GaN HEMT small-signal model according to the embodiment of the inventiongsS in S parameter of rear pair small signal11The improvement of (2) is compared with the schematic diagram. As shown in fig. 6It can be seen that if R is not presentgsThen gate resistance RgWill be too large, S11The offset is generated at high frequency, the characteristic of the device at high frequency is influenced, and R is introducedgsThen to RgIs more accurate, thus improving S at high frequencies11Curve fitting, where 'o' is the result of measuring S parameters and '-' is the result of simulating S parameters, shows that the right graph is significantly improved compared to the left graph.
FIG. 7 shows a drain delay factor τ by introducing a source-drain conductance in the parameter extraction method of the AlGaN/GaN HEMT small-signal model according to the embodiment of the present inventiondsS in S parameter of rear pair small signal22The improvement situation is compared with the schematic diagram. As shown in FIG. 7, τ is not introduceddsAt high frequency S22Is greatly influenced, and when extracting internal parameters, CdsNegative values of the parameters often occur. The time constant of the area under the gate not only comprises transconductance gmIs a delay factor τ ofmAnd also its drain-side delay factor tau caused by a large depletion region between the gate and drainds,τdsThe introduction of the method improves the delay of a drain end signal along with a gate end signal, and simultaneously more accurately describes the change of the signal transmission along with the bias voltage and the frequency. In the figure, 'o' is the result of measuring the S parameter, and '-' is the result of simulating the S parameter, so that the right figure is obviously improved relative to the left figure.
Table 1 shows all small-signal parameter values of the devices extracted by the parameter extraction method for the AlGaN/GaN HEMT small-signal model according to the embodiment of the present invention. Table 1 shows the parameters extracted for the device structure of fig. 1, which is a GaN HEMT device with a gate length of 0.3um and a gate width of 4 × 100 um.
TABLE 1
Cpg=9.96fF Cpg=13.04fF Cpgd=2.34fF Rg=4.08Ohm
Rd=0.5Ohm Rs=4.0Ohm Rgs=4.5Ohm Lg=0.02pH
Ld=0.04pH Ls=0.001pH Cgs=0.197pF Cgd=0.074pF
Cds=0.025pF gm=65.06mS τm=1.3ps Ri=1.5Ohm
gds=3.08mS τds=5.62ps
The traditional extraction of small signal model parameters often has the condition that the extraction parameters are negative, andthe corresponding S-parameter fit is not good. Aiming at the structure and the characteristics of the AlGaN/GaN HEMT device, because the small signal parameters are connected with the physical topological structure of the device, the connection with the process characteristics and the physical characteristics is tight, so the parameters needing to be extracted have physical significance, a specific standard is provided for the parameter extraction method, when the accuracy of the small signal parameters is measured, the small signal parameters are firstly ensured to be consistent with the physical parameter significance of the extracted device, the S parameter fitting effect is good, and simultaneously, the high-frequency characteristic parameter f can be accurately predictedtAnd fmaxTherefore, the method is significant for researching the extraction method of the small signal parameters.
The parameter extraction method of the AlGaN/GaN HEMT small-signal model mainly adopts the same peripheral structure of the device with the extracted parameters and adopts an open circuit de-embedding pattern without an active region part to calculate the peripheral parasitic capacitance of the device; introducing parasitic Schottky gate resistance RgsThe resistance is calculated by adopting a test under a specific condition, so that the extraction of the series parasitic resistance and the inductance is more reasonable, and the accurate intrinsic parameters are laid; introducing a delay factor tau in the source-drain conductancedsCan improve CdsThe extraction of (2) makes the description of the equivalent circuit of the device more reasonable. Through experimental verification, the whole parameter extraction process is used for reducing negative values and improving the fitting of S parameters, particularly S11And S22The improvement of the fitting effect of (2) is particularly significant.
The accurate extraction of small signal parameters has important significance for guiding process steps, detecting process accuracy, improving device structure, researching the influence of specific parameters on the high-frequency characteristics of the device, monitoring and comparing experimental results, and simultaneously lays a foundation for the establishment of a large signal model of the device, thereby having good research significance and practical value.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (6)

1. A parameter extraction method for an AlGaN/GaN HEMT small-signal model is characterized by comprising the following steps:
step 10: measuring scattering parameter S of peripheral open-circuit de-embedding circuit, converting to obtain admittance parameter Y, and calculating peripheral parasitic capacitance Cpg、CpdAnd CpgdThe peripheral open circuit de-embedding circuit comprises a peripheral parasitic capacitance Cpg、CpdAnd CpgdSaid peripheral parasitic capacitance CpgdSeries CpgAnd CpdTo (c) to (d);
step 20: at Vgs>0,VdsUnder the bias state of 0V, the gate voltage values of two groups of AlGaN/GaN HEMT devices are respectively selected to be Vgs1And Vgs2Testing to obtain two groups of current values I which respectively correspond to the two groups of gate voltage values and are less than 10mAgs1And Igs2The measured gate voltage values are respectively Vgs1And Vgs2The S parameter is converted to obtain an impedance parameter Z, and then the values of the series parasitic resistance and the series parasitic inductance are calculated;
step 30: measuring S parameter of AlGaN/GaN HEMT device in bias state, and removing peripheral parasitic capacitance C calculated in step 10pg、CpdAnd CpgdAnd the series parasitic resistance R obtained in step 20g、Rd、RsAnd series parasitic inductance Lg、Ld、LsObtaining intrinsic S parameters of the internal parameters, converting the intrinsic S parameters to obtain Y parameters, and calculating to obtain internal intrinsic parameter gate capacitance C in a bias stategs、CgdTransconductance gmAnd its delay factor taumSource drain conductance gdsAnd its drain terminal delay factor taudsDrain-source capacitance CdsAnd gate source channel resistance RiThe numerical value of (c).
2. The method for extracting parameters of an AlGaN/GaN HEMT small-signal model according to claim 1, wherein the peripheral parasitic capacitance C is calculated in the step 10pg、CpdAnd CpgdThe process of numerical values of (1) includes: converting the S parameter to obtain a Y parameter according to formulas (1) - (3)
Im(Y11)=ω(Cpg+Cpgd) (1)
Im(Y12)=Im(Y21)=-ωCpgd (2)
Im(Y22)=ω(Cpd+Cpgd) (3)
Calculating to obtain the peripheral parasitic capacitance Cpg、CpdAnd CpgdRespectively are
<math> <mrow> <msub> <mi>C</mi> <mi>pgd</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>,</mo> <msub> <mi>C</mi> <mi>pg</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>-</mo> <msub> <mi>C</mi> <mi>pgd</mi> </msub> <mo>,</mo> <msub> <mi>C</mi> <mi>pd</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>-</mo> <msub> <mi>C</mi> <mi>pgd</mi> </msub> <mo>.</mo> </mrow> </math>
3. The method for extracting parameters of a small signal model of AlGaN/GaN HEMT according to claim 1, wherein the gate voltage values of the two groups of AlGaN/GaN HEMTs selected in the step 20 are both greater than the schottky voltage of the AlGaN/GaN HEMT, and the current values corresponding to the gate voltage values of the two groups of AlGaN/GaN HEMTs are both less than 10 mA.
4. The method of claim 3, wherein the step 20 is performed at Vgs>0,VdsUnder the bias state of 0V, the equivalent circuit of the AlGaN/GaN HEMT device comprises a series parasitic resistor Rg、RdAnd RsSeries parasitic inductance Lg、LdAnd LsMultiple gate distributed resistance Δ RgsMultiple gate terminal distributed capacitance Δ CgMultiple channel distributed resistance Δ RchAnd a plurality of channel distributed capacitances Δ CdsWherein the gate terminal has a distributed resistance Δ RgsAnd gate terminal distributed capacitance Δ CgAre connected in parallel to form a grid end distributed parallel unit, and a plurality of grid end distributed parallel units are connected in parallel with a series parasitic resistor RgAre connected in series, the series parasitic resistance RgAnd series parasitic inductance LgAre connected in series, and the channel distribution resistance is delta RchAnd channel distributed capacitance Δ CdsAre connected in parallel to form a channel distribution parallel unit, and one end of the plurality of channel distribution parallel units is connected with a series parasitic resistor R after being connected in seriesdAre connected in series, and the other end is connected with a series parasitic resistor RsAre connected in series, the series parasitic resistance RdAnd series parasitic inductance LdAre connected in series, the series parasitic resistance RsAnd series parasitic inductance LsAre connected in series.
5. The method of claim 3, wherein the step 20 is to extract V from the AlGaN/GaN HEMT small-signal modelgs1And Vgs2Two groups of Z parameters obtained by the transformation of the two groups of S parameters are respectivelyAnd
Figure FDA0000038361460000023
according to the formula
n = Re ( Z 11 1 ) - Re ( Z 11 2 ) kT / ( qI gs 1 - q I gs 2 ) And R gs = nkT qI gs
n and R are obtained by calculationgs,RgsThe schottky resistance at the gate end can be obtained according to the transmission line equivalent equation as follows:
<math> <mrow> <msub> <mi>Z</mi> <mn>11</mn> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>g</mi> </msub> <mo>+</mo> <mfrac> <msub> <mi>Z</mi> <mn>0</mn> </msub> <mrow> <mi>tanh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>j&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>L</mi> <mi>g</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </math>
<math> <mrow> <msub> <mi>Z</mi> <mn>12</mn> </msub> <mo>=</mo> <msub> <mi>Z</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>Z</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>cosh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sinh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>j&omega;</mi> <msub> <mi>L</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </math>
<math> <mrow> <msub> <mi>Z</mi> <mn>22</mn> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>R</mi> <mi>d</mi> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mrow> <mn>2</mn> <mi>Z</mi> </mrow> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>cosh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sinh</mi> <mrow> <mo>(</mo> <mi>&gamma;L</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>j&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>L</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </math>
wherein Δ Rch=Rch×Δx,ΔCds=Cds×Δx,ΔRgs=Rgs×Δx,ΔCg=CgX Δ x, Δ x is an infinitesimal length along the gate length direction.
Wherein: z0Is the characteristic impedance, γ L is the product of the transmission constant and the length along the gate length,
<math> <mrow> <msub> <mi>Z</mi> <mn>0</mn> </msub> <mo>=</mo> <msqrt> <mfrac> <mrow> <msub> <mi>R</mi> <mi>ch</mi> </msub> <mo>/</mo> <mo>/</mo> <msub> <mi>C</mi> <mi>ds</mi> </msub> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <msub> <mi>R</mi> <mi>gs</mi> </msub> <mo>+</mo> <mi>j&omega;</mi> <msub> <mi>C</mi> <mi>g</mi> </msub> </mrow> </mfrac> </msqrt> </mrow> </math>
<math> <mrow> <mi>&gamma;L</mi> <mo>=</mo> <msqrt> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>ch</mi> </msub> <mo>/</mo> <mo>/</mo> <msub> <mi>C</mi> <mi>ds</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>/</mo> <msub> <mi>R</mi> <mi>gs</mi> </msub> <mo>+</mo> <mi>j&omega;</mi> <msub> <mi>C</mi> <mi>g</mi> </msub> <mo>)</mo> </mrow> </msqrt> </mrow> </math>
since (gamma L)2|<<1,ωRchCds<<1,ωRgsCg1, the Z parameter forms of the formulas (7) to (9) can be obtained through simplification,
Z11=Rs+Rg+RgsgRch+jω(Ls+Lg) (7)
Z12=Z21=Rs+αRch+jωLs (8)
Z22=Rs+Rd+2αRch+jω(Ls+Ld) (9)
wherein due to Re (Z)11) And
R gs = nkT qI gs
proportional relation, so obtained by measurement
Figure FDA0000038361460000036
The intercept of the linear fit is Rs+RggRchAnd at the same time due to
Figure FDA0000038361460000037
Wherein VthIs measured as a threshold voltage
Figure FDA0000038361460000038
The fitting intercept of the low leakage pressure is Rd+RsThus, in combination with the above formulas (7) to (9), the corresponding resistance value can be calculated, and then the series parasitic resistance and the series parasitic inductance are obtained by calculating the real part and the imaginary part respectively:
Rg=real(Z11)-real(Z12)+Rch/6,Rd=real(Z22)-real(Z12)-Rch/2,
Rs=real(Z12)-Rch/2,Lg=imag(Z11-Z12)/ω,
Ld=imag(Z22-Z12)/ω,Ls=imag(Z12)/ω,
wherein alpha isgAnd alpha is typically 1/3 and 1/2.
6. The method for extracting parameters of AlGaN/GaN HEMT small-signal model according to claim 1, wherein the gate capacitance C of the intrinsic parameter value in the biased state is obtained by calculation in step 30gs、CgdTransconductance gmAnd its delay factor taumSource drain conductance gdsAnd its drain terminal delay factor taudsDrain-source capacitance CdsAnd gate source channel resistance RiThe process comprises the following steps: the intrinsic S parameter of the internal parameter is converted to obtain the Y parameter according to the formulas (10) to (13)
<math> <mrow> <msub> <mi>Y</mi> <mn>11</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>R</mi> <mi>i</mi> </msub> <msubsup> <mi>C</mi> <mi>gs</mi> <mn>2</mn> </msubsup> <msup> <mi>&omega;</mi> <mn>2</mn> </msup> </mrow> <mi>D</mi> </mfrac> <mo>+</mo> <mi>j&omega;</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>C</mi> <mi>gs</mi> </msub> <mi>D</mi> </mfrac> <mo>+</mo> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow> </math>
Y12=-jωCgd (11)
<math> <mrow> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>g</mi> <mi>m</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>j&omega;</mi> <msub> <mi>R</mi> <mi>i</mi> </msub> <msub> <mi>C</mi> <mi>gs</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow> </math>
<math> <mrow> <msub> <mi>Y</mi> <mn>22</mn> </msub> <mo>=</mo> <msub> <mi>g</mi> <mi>ds</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j&omega;</mi> <msub> <mi>&tau;</mi> <mi>ds</mi> </msub> </mrow> </msup> <mo>+</mo> <mi>j&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>ds</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow> </math>
Wherein, <math> <mrow> <mi>D</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>&omega;</mi> <mn>2</mn> </msup> <msubsup> <mi>C</mi> <mi>gs</mi> <mn>2</mn> </msubsup> <msubsup> <mi>R</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </math>
and calculating the values of the intrinsic parameters in the parts as g by respectively equalizing the real part and the imaginary partds=real(Y22), <math> <mrow> <msub> <mi>&tau;</mi> <mi>ds</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&omega;</mi> </mfrac> <msqrt> <mrow> <mo>(</mo> <mn>2</mn> <mo>-</mo> <mn>2</mn> <mo>*</mo> <mi>real</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>g</mi> <mi>ds</mi> </msub> <mo>)</mo> </mrow> </msqrt> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>C</mi> <mi>gs</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mi>&omega;</mi> </mfrac> <mo>*</mo> <mrow> <mo>(</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mi>D</mi> <mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>*</mo> <mrow> <mo>(</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>C</mi> <mi>gd</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>,</mo> <msub> <mi>C</mi> <mi>ds</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>22</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>&omega;</mi> </mfrac> <mo>+</mo> <msub> <mi>g</mi> <mi>ds</mi> </msub> <mo>*</mo> <msub> <mi>&tau;</mi> <mi>ds</mi> </msub> <mo>,</mo> </mrow> </math>
g m = ( 1 + D 2 ) * ( real ( Y 21 ) 2 + ( imag ( Y 21 ) - imag ( Y 12 ) ) 2 ) ,
<math> <mrow> <msub> <mi>&tau;</mi> <mi>m</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mi>&omega;</mi> </mfrac> <mo>*</mo> <mi>arctan</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>D</mi> <mo>*</mo> <mi>real</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>real</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>D</mi> <mo>*</mo> <mrow> <mo>(</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>imag</mi> <mrow> <mo>(</mo> <msub> <mi>Y</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </math>
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