GaN HEMTs with gate-connected FP based on Equivalent Potential Method

Journal Pre-proof Analytical model for the potential and electric field distributions of AlGaN/GaN HEMTs with gate-connected FP based on Equivalent Potential Method Jianhua Liu, Yufeng Guo, Jun Zhang, Jiafei Yao, Xiaoming Huang, Chenyang Huang, Zhi Huang, Kemeng Yang PII:

S0749-6036(19)31257-1

DOI:

https://doi.org/10.1016/j.spmi.2019.106327

Reference:

YSPMI 106327

To appear in:

Superlattices and Microstructures

Received Date: 17 July 2019 Revised Date:

25 October 2019

Accepted Date: 29 October 2019

Please cite this article as: J. Liu, Y. Guo, J. Zhang, J. Yao, X. Huang, C. Huang, Z. Huang, K. Yang, Analytical model for the potential and electric field distributions of AlGaN/GaN HEMTs with gateconnected FP based on Equivalent Potential Method, Superlattices and Microstructures (2019), doi: https://doi.org/10.1016/j.spmi.2019.106327. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Analytical model for the potential and electric field distributions of AlGaN/GaN HEMTs with gate-connected FP based on Equivalent Potential Method Jianhua Liu1,2, Yufeng Guo1,2, Jun Zhang1,2, Jiafei Yao1,2, Xiaoming Huang1,2, Chenyang Huang1,2, Zhi Huang1,2 and Kemeng Yang1,2 1

College of Electronic and Optical Engineering and College of Microelectronics, Nanjing University of Posts and Telecommunications, Nanjing 210023, China 2 National and Local Joint Engineering Laboratory of RF Integration and Micro-Assembly Technology, Nanjing University of Posts and Telecommunications, Nanjing 210023, China *E-mail: [email protected]

Abstract A new physics-based modeling approach, Equivalent Potential Method (EPM), is first proposed in this paper to simplify the modeling progress of AlGaN/GaN HEMTs. The EPM indicates that charges in depletion region can be equivalent to the potential at passivation surface layer. Hereby, the depletion region can be seen as neutral, which therefore can be analyzed by using Laplace Equations other than the inherent Poisson Equations leading to much less modeling complexity. Especially, the corresponding iteration calculations can be significantly simplified with the elimination of various charge quantity. EPM provides an effective way to solve electric field and potential distributions with great concision and accuracy, especially important for AlGaN/GaN HEMTs with various charge layers to be concerned. By applying EPM, the analytical model shows good agreements with numerical results with acceptable discrepancies. The proposed simple analytical approach also offers effective guidance for field plates designing. Keywords: AlGaN/GaN HEMTs, Equivalent Potential Method (EPM), Analytical model, Field plate (FP), Electric field (E-field) distribution, Potential distribution.

Introduction Wide-Bandgap GaN-based high electron mobility transistors (HEMTs) are attracting increasing attention for their promising performance in high frequency, high power and high temperature applications [1-3]. These advantages originate from GaN material properties, including wide bandgap, low dielectric constant, high thermal conductivity and high electron saturation velocity. Besides these, high electron

mobility and high two-dimension electron gas (2-DEG) density induced by polarization effect also attribute to AlGaN/GaN HEMTs’ great performance [4]. Owing to its wide bandgap, the majority of efforts have been pursued on power applications. Breakdown voltage (BV) plays an important role in AlGaN/GaN HEMTs’ off-state performance [5-7]. It is generally recognized that the high Electric field (E-field) peak at the drain-side gate edge leads to AlGaN/GaN HEMTs’ breakdown. Characteristics of AlGaN/GaN HEMTs are mostly investigated by device fabrication with experiment tests and numerical simulation via TCAD software. Such approaches are very time-consuming and hard to provide physical insight into the device’s BV characteristic [8, 9]. Meanwhile, due to its distinctive material and device traits, the already mature analytical modeling techniques for silicon-based power devices are impractical to be used in GaN-based devices. Hence, great efforts have been devoted to develop a simple but accurate analytical model for AlGaN/GaN HEMTs. Since the existing modeling methods obtain analytical models via directly solving Poisson equations, deficiencies exist in these approaches which originate from its’ inevitable introduction of various charge quantity into the iterative calculation of a set of partial differential equations [10-12]. Meanwhile, AlGaN/GaN HEMTs contain various depletion charge layers as well as interface fixed charge layers thus making the solving of the Poisson equations becoming more complex. In this work, Equivalent Potential Method (EPM) is firstly proposed in AlGaN/GaN HEMTs to simplify the modeling process and provide effective designing guidance. By equivalenting both depletion charge layers and interface fixed charge layers into the potential at passivation surface, depletion region can be seen as neutral. Hence, Poisson equations can degenerate into Laplace equations. By using EPM, the modeling is significantly simplified. Based on the EPM, we proposed a simple but accurate analytical model of AlGaN/GaN MIS-HEMT with gate-connected field plate (FP). The veracity and simplicity of the proposed methodology are verified by the good agreement between analytical results and simulations obtained by Sentaurus, a commercial TCAD tool.

Model The schematic cross section of the recessed-gate AlGaN/GaN MIS-HEMT is shown in Fig. 1. AlGaN/GaN HEMTs with recessed-gate can operate in enhancement-mode (E-mode). Also, Metal-Insulator-Semiconductor (MIS) is widely used in AlGaN/GaN

HEMTs gate structure engineering for its capacity to achieve low gate leakage current. Hence, the AlGaN barrier layer is recessed and Si3N4 material is introduced as gate dielectric and passivation layer [13-15]. In addition, to mitigate the E-field peak at the drain-side gate edge, gate-connected FP is utilized to obtain more uniform E-field distribution and potential distribution. Corresponding parameters utilized are shown in Fig. 1. t1 and ε1 are the thickness and dielectric constant of the AlGaN barrier layer. t2 and ε2 are the thickness and dielectric constant of the GaN channel layer. tpas and εpas are the thickness and dielectric constant of the Si3N4 passivation layer. The donor concentration of AlGaN barrier layer is N1. The background donor concentration of GaN channel is N2. L1 denotes the distance from the drain-side gate edge to the FP edge. L2 denotes the distance from the drain-side gate edge to the boundary of depletion region obtained from numerical results. The physical models used in numerical simulations include basic Poisson, drift-diffusion and current-continuity equations. The physical parameter models contain Shockley-Read-Hall for recombination, impact ionization for generation, high field dependent mobility models, polarization model and carrier statistic model. The device is biased in “off-state” (VS=0V, VGS=0V, VDS=100 to 200V) to analyze the E-field distribution and potential distribution. Hereby, positive charge is introduced along the heterojunction interface at the AlGaN side to model the polarization effect, based on the experiment result of approximately neutral at the AlGaN/passivation layer interface [16].

LG

GND

LFP

VDS

tpas

Si3N4

S

G

εpass

ε1 N1 σ p AlGaN I II ++++++++++++++++ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2DEG ---GaN Channel III IV ε2 N2

D t1 x

t2

L1 L2

GaN Buffer y

Substrate Fig. 1. Schematic cross section of recessed gate AlGaN/GaN MIS-HEMT with

gate-connected FP.

The x-axis is established along the AlGaN/GaN heterojunction interface and the y-axis is established along the right side of gate dielectric, thus a two-dimension coordinate system is obtained. Based on numerous simulation results, the E-field peak at drain-side gate edge (x=0, y=0) or FP edge (x=L1, y=0), which are denoted as EG and

EFP respectively in this paper, leads to the breakdown of the device. Although

the depletion region could extend from the drain-side gate edge to the source, the E-field in this region is so low that it is neglectable. Hence, the AlGaN/GaN HEMT is divided into four regions when modeling, from the drain-side gate edge to the depletion region boundary, labeled: I, II, III, IV. The boundaries of each region are x=0, L1, L2; y=-t1, 0, t2. Compared with resultant E-field, the vertical E-field is neglectable. So the resultant E-field can be represented by the lateral E-field for simplicity. VDS

VG Si3N4 Passivation

VG

VDS

VG Si3N4 Passivation

dy1

dE1(y1)

y0

E0(y0)

dy2

dE2(y2)

GaN Buffer

GaN Buffer

Substrate

Substrate

(a)

(b)

Vequ=V0+V1+V2 Si3N4 Passivation

VDS

VG

dVequ(y)=V0+dV1(y1)+dV2(y2)

VDS

Si3N4 Passivation

E1

dE1(y1)

E0

E0(y0)

E2

dE2(y2)

GaN Buffer

GaN Buffer

Substrate

Substrate

(d)

(c)

Fig. 2. Charge distribution in depletion region and the progress of EPM.

As shown in Fig. 2(a), when biased in off-state, the charge in depletion region can be classified into three parts: ionized fixed positive charge in AlGaN layer, interface

fixed positive charge along heterojunction at AlGaN side and ionized fixed positive charge in GaN layer. As illustrated in Fig. 2(b), charge in depletion region can be divided into differential charge layers of y0, dy1 and dy2, respectively (y0=0, -t1 < y1< 0 and 0< y1< t2). Here, y0 is an inherent differential charge layer introduced to model the polarization effect. Moreover, the influence induced by differential charge layers can be evaluated as E-field denoted as dE1(y), E0(y) and dE2(y) respectively. In addition, the doping profile in every layer is uniform, so the doping concentration N (y1), N (y2) can be assumed as N1, N2 respectively. Hence, the charge quantity of differential charge layers yield as ݀ܳ଴ ሺ‫ݕ‬଴ ሻ = ‫ߪݍ‬௣ , ݀ܳଵ ሺ‫ݕ‬ଵ ሻ = ‫ܰݍ‬ଵ ሺ‫ݕ‬ଵ ሻ݀‫ݕ‬ଵ , ݀ܳଶ ሺ‫ݕ‬ଶ ሻ = ‫ܰݍ‬ଶ ሺ‫ݕ‬ଶ ሻ݀‫ݕ‬ଶ .The polarization charge density σp can be derived from Al composition x in the AlGaN material and the thickness of AlGaN layer, based on polarization effect theory given by ߪ୮ = |ܲ୔୉ ሺAl௫ Gaଵି௫ Nሻ + ܲୗ୔ ሺAl௫ Gaଵି௫ Nሻ − ܲୗ୔ ሺGaNሻ|. As shown in Fig2. (c), differential charge layers and passivation layer surface can be evaluated as plates of differential capacitance. The value of the differential capacitance can be evaluated as capacitance of different layers connected in series yielding as:

 t pas t1  C ( y0 ) =  +  ε   pas ε1 

−1

 t pas y1 + t1  C ( y1 ) =  +  ε ε1   pas

−1

 t pas t1 y2  C ( y2 ) =  + +  ε   pas ε1 ε 2 

−1

(1)

By using parallel plate capacitance theory, the differential charge layers can be equivalent to the differential potential applied at passivation layer surface. Thus, the same E-field can be induced by the equivalent differential potential as the differential charge layers can. Since the differential equivalent potential has been derived by dV = dQ / C , the total equivalent potential at passivation layer surface can be

obtained by integrating the differential equivalent potential as shown in Fig. 2(d).

t t V0 = qσ p  1 + pas  ε 1 ε pas 

  

 y +t t V1 = ∫ qN1 ( y1 )  1 1 + pas  ε1 − t1 ε pas  0

(2)

  t12 t1t pas +  dy1 = qN1    2ε 1 ε pas

 y t t2 t V2 = − ∫ qN 2 ( y )  + 1 + pas  ε 2 ε1 ε pas 0 

Vequ = V0 + V1 + V2

  

  t2 2 t1t2 t pas t2  dy = − qN + +  2 2  2ε 2 ε1 ε pas    

(3)

(4) (5)

Thus, through EPM, all charges in depletion region are transferred to the potential

at passivation layer surface which can be utilized as boundary conditions later. So the depletion region can be seen as neutral semiconductor. And the elimination of charge quantity leads to significant simplicity when modeling. When modeling, the two-dimension potential function in the four regions mentioned before can be denoted as φ1, 1 (x, y) (in Region I, 0
∂ 2ϕi , j ( x, y ) ∂x 2

+

∂ 2ϕi , j ( x, y ) ∂y 2

=0

i = 1, 2，j = 1, 2

(6)

The potential at the heterojunction interface is ϕ j ( x ) = ϕ1, j ( x, 0 ) = ϕ 2, j ( x, 0 ) ( j = 1, 2 ). Due to the depletion condition in Region I-IV, the channel resistance is apparently higher than the access region, leading VDS drop across the depletion region. Thus, heterojunction potential is assumed to be VDS at the right boundary of Region II. Moreover, since the spacing between the gate and channel is so thin (10nm) that the potential drop across the gate dielectric is neglectable. Hence, the heterojunction potential at the left boundary of Region I can be evaluated as VG.

ϕ1 ( 0 ) = VG , ϕ 2 ( L2 ) = VDS

(7)

The electric displacement and potential at AlGaN/GaN heterojunction are continuous in lateral, which yields: ∂ϕ1 ( x ) ∂ϕ ( x ) |x = L1 = 2 |x = L1 ∂x ∂x

, ϕ1 ( L1 , 0 ) =ϕ2 ( L1 , 0 )

(8)

According to electric displacement continuous at AlGaN/Si3N4 passivation interface, boundary condition yielding as: −ε 1

∂ϕ1, j ( x, y ) ∂y

|x =− t1 = ε pas

V f ( x ) + Vequ − ϕ1, j ( x, −t1 ) t pas

j = 1, 2

(9)

Vf (x) is the potential at the surface of Si3N4 passivation layer. By using EPM, the charge quantity at the heterojunction interface is eliminated and the depletion region can be equivalent as a neutral region. So the electric displacement continuous in vertical leads to formula (10).

−ε1

∂ϕ1, j ( x, y ) ∂y

| y =0 = −ε 2

∂ϕ2, j ( x, y ) ∂y

| y =0

j = 1, 2

(10)

The potential at the interface of AlGaN/GaN heterojunction is continuous in vertical leading to:

ϕ1, j ( x, 0 ) = ϕ2, j ( x, 0 )

j = 1, 2

(11)

According to simulation results, the vertical E-field at the bottom of GaN channel layer is so small that it can be evaluated as 0, especially when it is thicker than 0.2 um.

∂ϕ2, j ( x, y ) ∂y

| y =t2 = 0

j = 1, 2

(12)

In addition, equation (13) can be obtained according to the concentration continuous at heterojunction in vertical direction after employing EPM.

∂ 2ϕ1, j ( x, y ) ∂y 2

| y =0 =

∂ 2ϕ2, j ( x, y ) ∂y 2

| y =0

j = 1, 2

(13)

The potential function ϕi , j ( x, y) can be approximately simplified as parabolic functions shown in (14) in vertical direction based on Taylor’s formula.

ϕi , j ( x, y ) = ai , j + bi , j x + ci , j x 2

i = 1, 2 , j = 1, 2

(14)

So far, the boundary conditions have been obtained via equation (9) to (13). By submitting boundary conditions and equation (14) into Laplace equations (6), partial difference function can be derived as ∂ 2ϕ j ( x ) ϕ j ( x ) V f ( x ) + Vequ − 2 =− 2 ∂x t t2

j = 1, 2

(15)

Where t is the characteristic thickness of the device determined by device structure parameters, which can be expressed as 2 0.5 t=(t1 /2+ε2t1t2/ε1+ε1t1tpas/εpas+ε2t2tpas/εpas) . Here, equation (15) indicates that elements determining potential distribution conclude three components: structure parameters, charge in depletion region and applied VDS, which determine t, Vequ and Vf(x) respectively. Apparently, the Vf(x) in Region I and II is different. The potential of Vf(x) in Region I (0
ܸ௙,ଶ ሺ‫ݔ‬ሻ = ܸ஽ௌ , ‫ܮ‬ଵ < ‫ܮ < ݔ‬ଶ .In general, Vf (x) (L1
combination V f ,1 ( x) and V f ,2 ( x) by a coefficient η determined by device structure parameters. Based on the hypotheses above, the expressions of potential ϕ j ( x) ( j = 1, 2 ) can be denoted as ϕh1, j ( x) and ϕh1, j ( x) respectively.

x t

ϕh1,1 ( x,0 ) = A1 sinh + A2 sinh ϕh1,2 ( x,0 ) = B1 sinh

x − L1 L −x + B2 sinh 2 + V f ,1 ( x ) + Vequ t t x t

ϕh 2,1 ( x,0 ) = C1 sinh + C2 sinh ϕh 2,2 ( x,0 ) = D1 sinh

L1 − x + Vequ t

L1 − x + Vequ t

x − L1 L −x + D2 sinh 2 + V f ,2 ( x ) + Vequ t t

0 < x < L1

(16)

L1 < x < L2

(17)

0 < x < L1

(18)

L1 < x < L2

(19) By substituting (7), (8) into (16) - (19), coefficients used above can be solved out. V − Vequ Vequ tVDS + GS − L −L L2 − L1 sinh L1 sinh 2 1 t t A1 = , L1 L1 L2 − L1 cosh + sinh coth t t t

A2 =

VGS − Vequ , L1 sinh t

B1 = −

Vequ , L2 − L1 sinh t

V − Vequ Vequ tVDS + GS − L L −L L2 − L1 sinh 1 sinh 2 1 t t B2 = L2 − L1 L1 L2 − L1 sinh coth + cosh t t t    VGS − Vequ  Vequ L L L −L − tanh 1 + VDS tanh 1 coth 2 1   L L −L t t t  sinh 1 sinh 2 1  t t  , C1 =  L1 L L L −L sinh + tanh 1 sinh 1 cosh 2 1 t t t t

(20)

C2 =

VGS − Vequ , L1 sinh t

D1 = −

Vequ , L2 − L1 sinh t

   VGS − Vequ  Vequ L − tanh 1 − VDS  L1 L2 − L1  t  sinh  sinh t t  D2 =  L2 − L1 L L −L + tanh 1 cosh 2 1 sinh t t t

(21)

Hence, by introducing the weight coefficient η mentioned before, potential

ϕ j ( x) ( j = 1, 2 ) yield as:

ϕ1 ( x ) = (η A1 + (1 − η ) C1 ) sinh + (η A2 + (1 −η ) C2 ) sinh x t

L1 − x + Vequ t

0 < x < L1

(22)

ϕ 2 ( x ) = (η B1 + (1 − η ) D1 ) sinh +Vequ + VDS −

ηVDS ( L2 − x )

x − L1 L −x + (η B2 + (1 − η ) D2 ) sinh 2 t t

L2 − L1

L1 < x < L2

(23)

Furthermore, E-field distribution at the AlGaN/GaN heterojunction interface can be obtained by differential calculation.

E1 ( x ) =

(η A + (1 −η ) C ) cosh x − (η A + (1 −η ) C ) cosh L − x

E2 ( x ) =

(η B + (1 − η ) D ) cosh x − L − (η B + (1 − η ) D ) cosh L

1

1

t

1

2

t

L1 < x < L2

1

t

0 < x < L1 (24)

1

t

1

t

2

t

2

2

t

2

− x ηVDS + t L2 − L1

(25)

Results and discussion To verify the analytical model, numerical and analytical results are shown in Fig. 3, which show a fair accordance with some acceptable discrepancies. The lateral E-field distribution of heterojunction interface from drain-side gate edge to FP edge

presenting the trend of an exponential decrease from EG followed by an exponential increase to EFP. After that is a decline from EFP to nearly zero, which fits the E-field distribution in partial depletion region. Owing to the boundary condition Vf(x) is simplified as a combination of two assumptions, some discrepancies appear in the partial depletion region. Yet this is acceptable for two reasons. Firstly, analytical and numerical results have almost the same trend from EFP to nearly zero. Secondly, the electric characteristic almost determined by E-field peaks other than a specific decline in partial depletion region. In addition, slight discrepancies can be seen at the drain-side gate edge owing to neglecting the curvature effect in 2D modeling. simulation 100V 150V 200V analytical 100V 150V 200V

2.0

1.5

200

Potential (V)

Electric Field (MV/cm)

2.5

1.0

150

simulation 100V 150V 200V analytical 100V 150V 200V

100

50

0.5

0.0

0

3.5

4.0

4.5

W (µm)

5.0

5.5

6.0

3.5

4.0

4.5

5.0

5.5

6.0

W (µm)

(a) (b) Fig. 3. E-field distribution (a) and Potential distribution (b) of different VDS. W denotes the distance to the left boundary of the device, which is the same in Fig. 4, Fig. 5 and Fig. 6.

The feature of E-field and potential distribution for varying applied VDS are illustrated in Fig. 3. Device structure parameters are fixed and VDS set as 100V, 150V and 200V respectively. EFP increases with improving VDS, while EG nearly keeps unchanged. At the same time, the partial depletion region keeps extending and undertakes most of the increasing VDS, with the E-field and potential under the FP almost unchanged. This means the FP with specific bias acts as a role to maintain the electric characteristic under FP unchanging and leads the partial depletion region holding increasing VDS. In other words, once the FP structure is determined, the potential can be undertaken by the channel under the FP is almost determined. So, optimizing the FP to improve its ability to undertake more applied potential is promising.

simulation VG=0V VG=-4V VG=-8V analytical VG=0V VG=-4V VG=-8V

2.0

1.5

200

150

Potential (V)

Electric field (MV/cm)

2.5

simulation VG=0V VG=-4V VG=-8V analytical VG=0V VG=-4V VG=-8V

100

1.0

50

0.5 0

0.0 3.5

4.0

4.5

5.0

5.5

6.0

3.5

4.0

4.5

W (µm)

5.0

5.5

6.0

W (µm)

(a) (b) Fig. 4. E-field distribution (a) and Potential distribution (b) of the device with different gate bias.

In practice, gate bias usually ranges from -10V to 0V when AlGaN/GaN HEMTs operate in off-state, while the drain electrode is usually applied with serval hundreds of volts. As shown in Fig.4, since the gate bias is relatively small comparing the drain bias, the gate bias variation has a minor impact on the potential and electric field distribution. simulation 50nm 100nm 200nm analycal 50nm 100nm 200nm

Electric field (MV/cm)

3.0 2.5 2.0 1.5

200

Potential (V)

3.5

150

simulation 50nm 100nm 200nm analycal 50nm 100nm 200nm

100

1.0 50

0.5 0.0

3.5

4.0

4.5

W (µm)

5.0

5.5

0 3.5

4.0

4.5

5.0

5.5

W (µm)

(a) (b) Fig. 5. E-field distribution (a) and Potential distribution (b) of the device with different passivation layer thickness.

To evaluate the impact caused by thickness variation of passivation layer, device is biased at VDS (200V). Then analytical and numerical results with passivation layer thickness chosen as 50nm, 100nm and 200nm respectively, are verified in Fig. 5. When passivation layer thickness decrease, the EG decreases and EFP increases with a much higher rate. At the same time, the maximum E-field peak switched from EG to EFP. So, equal EG and EFP can be obtained with optimization passivation layer thickness, to achieve more uniform E-field distribution. This is attractive because with maximum E-field peak less than EC (Critical E-field, 3MV for GaN), more uniform

E-field distribution attributes to higher BV. Fig. 6 shows the analytical and numerical results of different FP lengths of 1um, 1.5um and 2um respectively. EG slightly decreases and maintains at a saturation value while EFP almost unchanging with increasing FP length. In addition, with FP extending, the E-field at the middle of the FP nearly reaches zero, almost undertaking no potential. Moreover, from Fig. 6(b), the potential undertaken by the region under FP almost is unchanged with the increase of FP length. So, with other parameters fixed, engineering FP length too long is meaningless and an optimization FP length can be derived. 3.5

Electric field (MV/cm)

2.5 2.0 1.5

200

Potential (V)

simulation 1um 1.5um 2um analytical 1um 1.5um 2un

3.0

150

simulation 1um 1.5um 2um analytical 1um 1.5um 2um

100

1.0 50

0.5 0

0.0 3.5

4.0

4.5

5.0

5.5

6.0

6.5

W (µm)

3.5

4.0

4.5

5.0

5.5

6.0

6.5

W (µm)

(a)

(b) Fig. 6. E-field distribution (a) and Potential distribution (b) of the device with different FP length.

Conclusion In order to provide an effective analysis methodology on GaN HEMTs with both simplicity and veracity, the Equivalent Potential Method (EPM) is firstly proposed in this paper to transfer the influence induced by the charge in depletion region to the potential at passivation layer surface as boundary conditions. By applying the proposed method, 2-D Poisson equations are degenerated to simple Laplace equations, with depletion region being equivalent to a neutral semiconductor. Hence, the iteration calculation can be significantly simplified with the elimination of various charge quantity. By applying EPM, regardless of the complexity of charge layers that may have, devices can always be equivalent to two neutral layers along with heterojunction interface with potential at passivation surface as boundary conditions. The well-agreed analytical and simulation results show great veracity and simplicity and offer effective guidance for FP engineering.

Acknowledgments This work was supported in part by the China Postdoctoral Science Foundation under Grant 2018M642291, National Natural Science Foundation of China under Grant 61574081, Grant 61704084 and Grant 61874059.

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