Simplified EPFL GaN HEMT Model
Farzan Jazaeri Majid Shalchian Ashkhen Yesayan Amin Rassekh Anurag Mangla Bertrand Parvais Jean-Michel Sallese







Simplified EPFL GaN HEMT Model

This project is funded by the Swiss National Science Foundation - project 200021 213116.

    Farzan Jazaeri, Majid Shalchian, Ashkhen Yesayan, Amin Rassekh, Anurag Mangla, 
 Bertrand Parvais, and Jean-Michel Sallese
Farzan Jazaeri, Ashkhen Yesayan, and Jean-Michel Sallese are with the Electron Device Modeling and Technology Laboratory (EDLAB) of the École Polytechnique Fédérale de Lausanne (EPFL), Switzerland (e-mail:farzan.jazaeri@epfl.ch).Majid Shalchian is with the department of Electrical Engineering, Amirkabir University of Technology. Amin Rassekh is with InCize, Louvain-la-Neuve, Belgium. Anurag Mangla is an alumnus of EPFL. 
Bertrand Parvais is affiliated with IMEC in Leuven and holds a position as a Guest Professor at Vrije Universiteit Brussels, Belgium.

    September 9, 2024
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§ ABSTRACT 
This paper introduces a simplified and design-oriented version of the EPFL HEMT model<cit.>, focusing on the normalized transconductance-to-current characteristic (Gm/I_D). Relying on these figures, insights into GaN HEMT modeling in relation to technology offers a comprehensive understanding of the device behavior. Validation is achieved through measured transfer characteristics of GaN HEMTs fabricated at IMEC on a broad range of biases. 
This simplified approach  should enable a simple and  effective circuit design methodology with AlGaN/GaN HEMT heterostructures.



§ INTRODUCTION



High Electron Mobility Transistors (HEMTs) have garnered interest due to their exceptional electron transport capabilities boosted by a quantum well that enables high-speed and high-power applications in microwave and millimeter-wave domains.

Despite these advantages, existing compact models and traditional design methods with HEMTs rely on outdated interpretations of their behavior which limits hand calculations in a preliminary design phase. To this purpose, we introduce the concept of inversion coefficient introduced initially in MOSFETs <cit.>, combined with the important G_m/I_D figure of merit. This approach leads to a comprehensive set of analytical expressions based on the charge-voltage relationships outlined in <cit.> and valid in all the regions of operation. Likewise for the MOSFET, we aim at giving a simple and effective design-oriented modelling framework.

The simplified model relies only on six model parameters:
the slope factor n_q, the threshold voltage V_T0, the specific current I_sp, the velocity saturation coefficient λ_c, the mobility reduction coefficient θ, and the charge threshold voltage, V_T. This leads to a general modeling and designing methods with HEMTs since the model inherits from the regular silicon MOSFET features, but with new physical quantities, a kind of generalized MOSFET. Then, the design strategy adopted in CMOS circuits can be seamlessly applied to the design of circuits based on HEMTs.



§ DESIGN-ORIENTED MODELING IN HEMTS

To extract analog design parameters that are used to predict the electrical characteristics of HEMTs, we propose to simplify the EPFL HEMT model making use of the G_m/I_D methodology. This model relies on semiconductor physics, making it a valuable tool for optimizing circuit designs with GaN with few model parameters. 




 §.§ Charge-Voltage Dependence

The typical structure of a HEMT is shown in Fig. 1. It consists in a large bandgap AlGaN semiconductor playing the role of the confinement barrier combined with a smaller bandgap GaN semiconductor.
For simplicity we propose to reuse the explicit and continuous charge-based relationship developed in <cit.> that we recal hereafter

    U_Tln[exp(n_ch/DoS_2D U_T)-1]+qn_ch/C_b n_q  +u_1√(n_ch^2)=ψ_p-V,

where V= U_Tln(N_A/N_V)+E_g/q+V_ch, u_1=Γ/q, C_b=-ε_1/x_1 where x_1 is the AlGaN layer thickness, U_T is the thermal voltage (k_BT/q), E_g is the band gap of GaN and the temperature-dependency of E_g is described in <cit.>:

    E_g(T)=E_g(0)-5.08× 10^-4 T^2/(996-T),

where E_g (0) = 3.28eV for Zinc Blende crystal structure, and other symbols are defined in Table <ref>.
As initially introduced in <cit.> for MOSFETs, then adopted in HEMTs <cit.>, the pinch-off surface potential ψ_p is defined as the surface potential ψ_s when n_ch=0  

    Q_ch=-qn_ch=C_b(V_GB+V_A+ψ_s+γ√(ψ_s) ),

leading to

    ψ_p=V_GB-V_A-γ^2(√(V_GB-V_A/γ^2+1/4)-1/2)

where the body factor γ and the voltage offset V_A are given by

    γ=-x_1/ε_1√(2ε_2N_Aq),


    V_A=-Δ E_C/q+ϕ_B,eff-qN_D/2ε_1x_1^2.


The effective Schottky barrier height ϕ_B,eff  accounts for polarization effects. Its value is given by ϕ_B_eff=ϕ_B+x_1σ_pol/ε_1 where σ_pol =σ_pol(GaN)- σ_pol(AlGaN).  The terms σ_pol(GaN) and σ_pol(AlGaN) are the polarization-induced charge surface densities localized at the GaN-AlGaN interface. The slope factor n_q is defined as the derivative of n_ch versus ψ_s, obtained from relation <ref>, evaluated at ψ_p (<cit.>),  leading to
n_q=1+γ/2√(ψ_p). This slope factor is part of the linearization scheme that links the mobile charge density to the surface potential (<cit.>, <cit.>):

    Q_ch=-n_q C_b (ψ_p-ψ_s),

where C_b holds for an equivalent gate stack capacitance. We can also use the first order term of the Taylor series of (<ref>) with respect to V_GB to get an approximate value for ψ_p, ψ_p≈ (V_GB-V_A)/n_q (<ref>). Replacing this value in (<ref>) leads to the following  normalized charge-based expression  

    ln[exp(α q_ch)-1]+2q_ch  + β√(q_ch^2)=V_GB-VT_0/n_qU_T,

where α =-Q_sp/qDoS_2DU_T, β=(U_1/U_T)√((-Q_sp/q)^2), Q_sp is the specific charge density defined as  Q_sp = -2n_qC_bU_T and V_T0  is the threshold voltage defined as  

    V_T0=V_A+n_qU_Tln(N_A/N_V)+n_qE_g/q+n_qV_ch.



 §.§ Drain to Source Current Derivation in HEMTs

The drain current expression of the EPFL HEMT model is given by I_ds=I_sp× i where i is the normalized current and I_sp is the so-called specific current defined as I_sp = 2n_qμ_0 C_bU^2_T W/L_G. Here, i is obtained by 

    i =(q^2_s+q_s-q^2_d+q_d)=i_f-i_r,

where  q_s =-q n_ch,s/Q_sp and q_d=-q n_ch,d/Q_sp are the normalized charge densities at source and drain, and i_f and i_r are defined as the forward and reverse normalized currents. Each of them is related to the normalized charge densities q_s and q_d by i_f,r =q_f,r^2+q_f,r.
These expressions make the HEMT model close the MOSFET <cit.> and motivate their use for circuit design. 



 §.§ Velocity Saturation and Mobility Reduction

Reducing the length of the transistor needs to include more features, such as mobility reduction, velocity saturation, and velocity overshoot for instance.
The mobility μ_eff = v_drift/E_y , called the effective mobility, links the drift velocity to the longitudinal and vertical electric fields. AlGaN semiconductors exhibit saturation of the drift velocity
(v_drift) for electrons (v_sat) when
the longitudinal electrical field E_y exceeds a
critical value E_crit, that also depends
on the vertical electrical field. In the effective mobility, normalizing the v_drift to v_sat and E_y to the E_crit leads to an effective mobility u_eff <cit.>

    u_eff=v_drift/v_sat/E_y/E_crit=μ_eff/μ_x=v/e

where e stand for the normalized electric field  (e = E_y/E_crit) and
v for the normalized velocity (v = v_drift/v_sat). The mobility μ_x is defined as μ_x= v_sat/E_crit and includes the effect of the vertical electrical field since the E_crit is also dependent on the vertical electrical field E_x.
The normalized effective mobility to the low-field mobility μ_0 is rewritten as 

    u=μ_eff/μ_0=μ_eff/μ_xμ_x/μ_0=u_eff× u_0,

where u_0 is given by

    u_0=μ_x/μ_0=[1+θ(V_GB-V_T)]^-1,

Note that in (<ref>), V_T is the charge threshold voltage which is not to be confused with the extrapolated threshold voltage, i.e. V_T0, and θ  corresponds to  the mobility reduction coefficient <cit.>. As already discussed in <cit.>, the drain current in HEMT is given by the drift-diffusion equation 
I_DS=μ_eff W [Q_chE_y+U_TdQ_ch/dy].

Normalizing the longitudinal electrical field, E(y), charge density, and current to E_crit, Q_sp, and I_sp respectively,  the current can be written in the normalized from as 

    i=-u (2q_che/λ_c+dq_ch/dξ),

where ξ=y/L_G, e=E_y/E_crit and λ_c, called the velocity saturation parameter, is defined as λ_c =2U_T/E_cL_G.
In a device operating in velocity saturated regime, the channel charge density at the drain approaches a saturated value q_d,sat which remains almost constant over the velocity saturated region, therefore dq_ch/dξ =0. Relying on the current continuity, using piecewise linear velocity-field model (i.e. |e|u_eff=1 for |e|≥ 1 and u_eff=u_0 for |e|<1), and integrating (<ref>) over the velocity saturated part of the channel close to the drain, the normalized drain current in a velocity saturated device is given by i_d,sat=2u_0q_d,sat/λ_c. On the other hand, relying on the current continuity along the channel, the drain current can be obtained from (<ref>) as<cit.>

    i_d,sat=u_0(q^2_s+q_s-q^2_d,sat-q_d,sat)=IC.

For the sake of simplicity, we first propose to neglect the impact of the mobility reduction due to the vertical electric field, assuming u_0= 1. Later on, this non-ideal effect will be merged into the solution of the input characteristic. Introducing inversion coefficient IC to determine the channel inversion level of a HEMT as IC= I_D,sat/I_sp =i_d,sat, the normalized source charge density in the saturation regime for a velocity saturated device is obtained form (<ref>):  

    q_s=-qn_ch|_s/Q_sp=1/2√(λ^2_cIC^2+2λ_cIC+4IC+1)-1/2.

It is worth mentioning that to some extent, naming 'inversion' and introducing IC to identify such a level of inversion in HEMT devices may seem unappropriated. However, to make HEMT look like a generalized silicon MOSFET we can still classify the operation regions of a HEMT as weak inversion (WI) for IC ≤ 0.1, moderate inversion (MI) for 0.1 < IC ≤ 10, and strong inversion (SI) for IC > 10, even though the channel is not resulting from an inversion process. 

Inserting relation (<ref>) into (<ref>) while imposing V_ch=0 leads to a  relationship between IC and V_GB in saturation mode. The relations (<ref>) and (<ref>) give an explicit solution of V_GB for a given IC. The general current–voltage relationship for a HEMT in saturation is independent of technological parameters and makes use of normalized variables. In the presence of mobility reduction due to the vertical field, IC can be replaced by u_0IC. Then using (<ref>), the large- and small-signal characteristics over a wide range of IC, from weak to strong 'inversion', can be fully captured by only six design parameters i.e. V_T0, n_q, I_sp, λ_c, V_T, and θ.
On the other hand, differentiating (<ref>), the input transconductance in a HEMT is obtained 

    g_m=∂ I_DS/∂ V_GB=I_sp[(2q_s+1)∂ q_s/∂ V_GB-(2q_d+1)∂ q_d/∂ V_GB]
 
where ∂ q_s/∂ V_GB and ∂ q_d/∂ V_GB are obtained from (<ref>) and are expressed by ∂ q_s/∂ V_GB≈∂ q_d/∂ V_GB≈ 1/(2n_qU_T). Therefore, an approximate value of g_m is given by 

    g_m=I_sp(q_s-q_d)/(n_qU_T).

 Knowing g_m, the best approach of interpreting the IC is through the transconductance efficiency, i.e. g_m/I_d versus IC  which is a key figure of merit widely used in CMOS circuit design: 

    g_mn_qU_T/I_DS   =q_s-q_d/i=1/q_s+q_d+1
       =2/√((1+λ_cIC)^2+4IC)+1+λ_cIC.

Considering the relationship between the transconductance efficiency and IC allows for a proper evaluation of the design trade offs among gain, performance of analog design, transistor dimensions. This ratio reach the maximum in the weak 'inversion' (g_mn_qU_T/I_DS = 1) where the drain current dependence is exponential versus V_GB. However, moving toward to the strong inversion, the g_mn_qU_T/I_DS dependence versus IC becomes linear and therefore g_mn_qU_T/I_DS≈ 1/(λ_cIC) for large values of IC. 


Once the g_m/I ratio is established, determining the optimal width-to-length ratio (W/L) of the transistor becomes straightforward. 





§ PARAMETER EXTRACTION PROCEDURE


This section describes a straightforward and precise method for parameter extraction for the simplified design-oriented EPFL HEMT model. This method necessitates six design parameters alongside some physical constants.

The initial step involves extracting n_q from the minimum of I_DS/g_mn_qU_T in weak inversion. It is worth noting that this ratio reaches the minimum value of 1 in weak inversion, leading to n_q=I_DS/g_mU_T. This minimum occurs when the drain current dependence is exponential with respect to V_GB.

Consequently, the slope factor can be extracted experimentally from the plot of I_DS/g_mU_T versus I_DS in weak inversion. Similarly, the specific current I_sp can be determined from the same plot. Adjusting I_sp to match the asymptotic curvature of n_q=I_DS/g_mU_T to the experimental data allows to extract I_sp.

Moving forward, λ_c is adjusted until the experimental data intersects the 1/λ_c IC line. Subsequently, the parameter n_q is modified to ensure that both the curves g_m.n.U_T/I_DS and n_q intersect the asymptotic lines simultaneously.

Then, V_T and V_T0 are utilized to fit the I_DS-V_GB curve near the threshold and the rising part of g_m. Finally, θ and other variables are fine-tuned to fit the I_DS-V_GB curve at higher V_GB and the falling part of the g_m curve.














§ RESULTS AND DISCUSSION

The model has been validated experimentally on GaN-on-Si HEMTs fabricated in IMEC following <cit.>. The epitaxial stack grown on 200 mm HR Si wafers by MOCVD used in this study is composed of: 5nm in-situ SiN passivation layer, 15nm AlGaN barrier, 1nm AlN spacer, 300nm GaN channel, on top of 1um C-doped back barrier and 1um transition layers.



In Figure 2, the slope factors (n_q) are derived from the plateau of I_D/(G_m·U_T) versus I_D in weak inversion (WI) for both long-channel (L_G = 3.0 μ m), shown in Fig. 2.a, and short-channel (L_G = 200 nm), depicted in Fig. 2.e. 
This essential extraction process captures the behavior of the drain current and transconductance, offering profound insights into device performance across different channel lengths.

Additionally, the normalized transconductance efficiency, G_mnU_T /I_D, is plotted versus the inversion coefficient IC. 
Figures 2.b and 2.f serve to corroborate the robustness of the G_m/I_D design methodology. In particular, from the bare intersection of the asymptotes in weak and strong 'inversion', see dashed lines in Figure 2, the key specific current parameter I_sp for both short and long channel devices is obtained.

Figures 2c and 2g present the I_D versus V_G characteristics for both long and short channel GaN HEMTs, depicted in both linear and logarithmic scales. These plots demonstrate that the simplified EPFL model accurately captures the input transfer behaviour across different channel lengths. Whether in the linear or logarithmic scale, the model accurately reflects the device characteristics, underscoring its efficacy in modelling both long and short channel devices. 

In summary, Figures 2d and 2h show a detailed comparison of the transconductance (G_m) against V_G for both long and short channel GaN HEMTs. The remarkable fidelity with which the simplified EPFL model captures both G_m and its peak values across varying gate voltages is confirmed. This robust agreement between model predictions and experimental data highlights the model's capability to accurately represent the transconductance behavior of GaN HEMTs. The extracted parameters for both long and short channel GaN HEMTs are listed in Table II. 
This validation underscores the simplified model effectiveness to model HEMTs using only a few parameters and enables a design and optimization of GaN-based devices in a more traditionnal way.


§ CONCLUSION

This paper introduces a simplified and design-oriented adaptation of the EPFL HEMT model, with a specific focus on the normalized transconductance-to-current characteristic  and IC. The research delves into GaN HEMT technology and modeling, aiming to provide a precise model for the electrical behavior of these devices using only a few parameters.
Validation is conducted by comparing measured transfer characteristics of GaN HEMTs at room temperature across a broad range of IC values. This study provides valuable guide in the effective design of circuits employing HEMTs.








































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