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A compact drain current model for heterostructure HEMTs including 2DEG density solution with two subbands
Solid-State Electronics 115 (2016) 54–59
Contents lists available at ScienceDirect
Solid-State Electronics journal homepage: www.elsevier.com/locate/sse
A compact drain current model for heterostructure HEMTs including 2DEG density solution with two subbands Wanling Deng a,b,⇑, Junkai Huang a, Xiaoyu Ma a, Juin J. Liou b, Fei Yu a a b
Department of Electronic Engineering, Jinan University, Guangzhou 510630, China Department of Electronic Engineering, University of Central Florida, Orlando 32826, USA
a r t i c l e
i n f o
Article history: Received 23 June 2015 Received in revised form 24 September 2015 Accepted 16 October 2015 Available online 11 November 2015 Keywords: High electron mobility transistors (HEMTs) Surface potential Fermi level Drain current model
a b s t r a c t An explicit and precise model for two dimensional electron gas (2DEG) charge density and Fermi level (Ef ) in heterostructure high electron mobility transistors (HEMTs) is developed. This model is from a con€dinger’s and Poisson’s equations in the quantum well with two important energy sistent solution of Schro levels. With these closed-form solutions, a unified surface potential calculation valid for all the operation regions is derived. With the help of surface potential, a single-piece drain current model is developed which is also capable of describing the current collapse effect by using a semi-empirical expression of source/drain access region resistances. Comparisons with numerical and measured data show that the proposed model gives an accurate description of Ef and drain current in all regions of operation. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, heterostructure high electron mobility transistors (HEMTs) have attracted much attention in high-speed and high-power applications. One of the most interesting properties of these devices is the formation of the two dimensional electron gas (2DEG) with a very high electron mobility at the heterointerface. From the computer-aided design perspective, there is still an urgent demand for an accurate and computationally efficient compact DC model for HEMTs. The formation of 2DEG in the quantum well near the heterointerface is the main principle of the HEMT device operation, and the modeling of 2DEG sheet carrier density (ns ) is a basic requirement in the development of a compact model for these devices. The calculation of ns has to be obtained from the well-known charge control equations [1] as a self-consistent solution of Poisson’s and €dinger’s equations in the quantum well. This quantum well Schro can be approximated as a triangular well and has two lowest subbands (E0 and E1 ). The major challenge in calculating ns is due to the complicated transcendental equation of ns which varies with the gate biases (V gs ). Obviously, the transcendental equation is not suitable for compact modeling. Recently, the physical-based approximations [2–4] for explicit solution of ns were developed and the surface-potential-based drain current models were also ⇑ Corresponding author at: Department of Electronic Engineering, Jinan University, Guangzhou 510630, China. Tel.: +86 020 85222481; fax: +86 020 85220231. E-mail address: [email protected] (W. Deng). http://dx.doi.org/10.1016/j.sse.2015.10.005 0038-1101/Ó 2015 Elsevier Ltd. All rights reserved.
presented. However, their calculations were derived by only considering one subband (E0 ) in the quantum well and ignoring the other one (E1 ). In our previous work [5], we also used this assumption and presented an explicit solution to efficiently compute ns and the quasi-fermi potential Ef . Compared with the models in [2–4], our scheme [5] is more straightforward and accurate. Nevertheless, as indicated in [6,7], the assumption of neglecting the contribution of the second subband (E1 ) is only valid for certain types of devices, and E1 has a significant impact on device characteristics when device parameters are varied. In [6], an initial result without considering E1 was obtained first. To enhance the accuracy of the initial, refinements including the contribution from E1 were carried out by Householder’s method for solving implicit functions. In addition, [6] does not take the case of Ef > E1 into account which is important for GaAs-based HEMTs. Therefore, Zhang et al. [7] improved the model in [6], and analytical expressions for ns and Ef as explicit functions of the terminal biases were proposed. However, this model is very complex. Thus, it is very important to achieve a simply but accurate solution for compact models. In this paper, accounting for the two lowest subbands in the quantum well, we present an improved analytical calculation for ns and Ef based on our previous work [5]. After that, the surface potential (ws ) calculation can be obtained using Ef . Therefore, a surface-potential-based compact model for HEMTs is developed to predict the current–voltage (I–V) characteristics. Furthermore, the current collapse in I–V characteristics is also captured by using a semi-empirical model for source/drain access region resistances.
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W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
2. The calculation of the Fermi level
where
b ¼ 3ðA=DÞV go ð2 þ A=DÞ2 ðc0 þ c1 Þ3 A2 ,
a ¼ ð2 þ A=DÞ3 ,
c ¼ 3ðA=DÞ V 2go ð2 þ A=DÞ þ 2ðc0 þ c1 Þ3 A2 V go , and e ¼ ðA=DÞ3 V 3go 2
€dinger’s As shown in Fig. 1, a self-consistent solution of Schro and Poisson’s equations in the triangular well including two subbands, is given by [7]
ns ¼
e qd
ðV gs V off /n Ef Þ
ð1Þ
ð2Þ
E0 ¼ c0 ns2=3
ð3Þ
E1 ¼ c1 ns2=3
ð4Þ
where e and d are the permittivity and thickness of the material between the gate and 2DEG, respectively, V off is the cutoff voltage, Ef is the Fermi potential with respect to the bottom of conduction band, /n is the channel potential, D is Density of States, c0 and c1 are determined by Robin boundary condition [8], and /th is the thermal voltage. Note that, Eqs. (1)–(4) are based on the one€dinger’s equations, but for dimensional (1D) Poisson’s and Schro high voltage and high field modes, the two-dimensional system has to be solved. For simplification, we only apply the 1D Poisson’s €dinger’s equations in this paper. and Schro As indicated by Zhang et al. [7], different sets of parameters fc0 ; c1 ; Dg lead to different contributions of E0 and E1 . Therefore, this variation results in a more complicated solution of Ef and ns . Obviously, as shown in Eqs. (1)–(4), the exact solutions of Ef and ns are transcendental in nature. To obtain a computationally efficient solution, which is suitable for circuit simulators, the regional approach [3] is employed here. As a result, three operation regions are divided, i.e., the strong 2DEG region, the moderate 2DEG region, and the subthreshold region.
1=3
b Y 1 Y 2 3a
ð8Þ
Y 1;2 ¼ A0 b þ 1:5aðB0
pffiffiffiffi DÞ
ð9Þ
where Ef ;1 denotes the Fermi potential in the case (A), D ¼ B20 2
4A0 C 0 ; A0 ¼ b 3ac; B0 ¼ bc 9ae, and C 0 ¼ c2 3be. For case (B), Eq. (2) is reduced to
ns ¼ DðEf E0 Þ ¼ DðEf c0 n2=3 s Þ:
ð10Þ
Eq. (10) is the same as the case where E1 is negligible, which can be solved in the same way as given before [5]. As a consequence, the solution for case (B) is marked as Ef ;0 . 2.2. Moderate 2DEG region In the moderate 2DEG region, we have 0 < Ef < E0 . Therefore, Eq. (2) can be simplified as [7]
! Ef c0 n2=3 s : /th
ns ¼ D/th exp
ð11Þ
It should be noted that, from [1], Eq. (2) can be re-expressed as
Ef ¼ /th lnðYÞ RþS þ Y¼ 2
ð12Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 RþS RSð1 ens =D/th Þ 2
between Ef and n2=3 using Eqs. (12) and (13) in Fig. 2, which shows s
Here, we assume that E1 > E0 . In the strong 2DEG region, we have Ef > E0 . Thus, there exist two possible conditions: (A) Ef > E1 and (B) E0 < Ef < E1 . For case (A), Eq. (2) can be approximated as
ð5Þ
Using Eq. (1), Ef as a function of V gs can be expressed as
2=3
a linear approximation between Ef and ns . Consequently, Eq. (11) can be approximated as
Ef c0 ðk3 Ef þ k1 Þ /th ð1 c0 k3 ÞEf c0 k1 ¼ D/th exp exp ; /th /th
ns ¼ D/th exp
ð6Þ
where A ¼ e=qd and V go ¼ V gs V off /n . Except for some coefficients, Eq. (6) is very similar to the relation of Ef vs. V gs in our previous work [5], but [5] neglected the contribution of E1 . Furthermore, Eq. (6) can be rewritten as a cubic equation 2
aE3f þ bEf þ cEf þ e ¼ 0
Fig. 1. Energy-band diagram of a HEMT.
ð13Þ
where R ¼ e1=/th E0 ; S ¼ e1=/th E1 , and Y ¼ e1=/th Ef . We plot the relation
2.1. Strong 2DEG region
AðV go Ef Þ ¼ 2DEf Dðc0 þ c1 ÞA2=3 ðV go Ef Þ2=3
1=3
Ef ;1 ¼
Ef E0 Ef E1 þ ln 1 þ exp ns ¼ D/th ln 1 þ exp /th /th
ns ¼ DðEf E0 Þ þ DðEf E1 Þ ¼ 2DEf Dðc0 þ c1 Þn2=3 s :
ðc0 þ c1 Þ3 A2 V 2go . The explicit solution of Ef can be calculated by the S. Fan formulas [9]
ð7Þ
Fig. 2. Ef as a function of n2=3 s .
ð14Þ
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W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
where k1 and k3 are fitting parameters, which can be extracted using two points in Fig. 2 to make a line. Substituting Eq. (1) into Eq. (14), we have
AðV go Ef Þ ¼ s exp
ð1 c0 k3 ÞEf /th
/th W0 1 c0 k3
sð1 c0 k3 Þ A/th
exp
V go /th =ð1 c0 k3 Þ
ð16Þ
where Ef ;mod means the value of Ef in the moderate 2DEG region. 2.3. Subthreshold region In the subthreshold region, Ef < 0, and therefore Ef E1 and Ef E0 become negative. The solution of the Fermi potential in this region (Ef ;sub ) is given by [7]
Ef ;sub ¼ V go /th W 0
2D V go : exp A /th
ð17Þ
2.4. Combining the regional models In the strong 2DEG region, the condition of Ef ¼ E1 determines the boundary between case (A) and case (B), at which the corresponding gate voltage is defined as V gs;E1 . When Ef ¼ E1 , Eq. (2) becomes
ns ¼ DðE1 E0 Þ þ D/th ln 2 ¼ Dðc1 c0 Þns2=3 þ D/th ln 2:
ð18Þ
Eq. (18) can be rewritten as a cubic equation, i.e.,
ðns MÞ3 ¼ n3 n2s
ð19Þ
where M ¼ D/th ln 2, and n ¼ Dðc1 c0 Þ. Herein, we still use the S. Fan formulas [9] to find an explicit solution
ns ¼
b1 Z 11=3 Z 1=3 2 3
Z 1;2 ¼ A1 b1 þ 1:5ðB1
ð20Þ pffiffiffiffi D1 Þ
ð21Þ 2
where b1 ¼ ð3M þ n3 Þ; c1 ¼ 3M2 ; e1 ¼ M3 , A1 ¼ b1 3c1 ; B1 ¼ B21
b1 c1 9e1 ; C 1 ¼ 3b1 e1 , and D1 ¼ 4A1 C 1 . As a consequence, from Eq. (1), V gs;E1 can be found as c21
V gs;E1 ¼
ns þ c1 ns2=3 þ V off þ /n : A
2=3
ð23Þ
A Taylor expansion to the first order is applied to further simplify Eq. (23). Combining with Eq. (2), yields
AV go ¼ D/th 1
c0 ðAV go Þ2=3 /th
#
:
pffiffiffi pffiffiffiffiffiffi A2 cos 3h þ 3 sin 3h 3a2
a2 ¼ r3 ; b2 ¼ A2 3jr2 ; c2 ¼ 3j2 r; e2 ¼ j3 ,
where
ð26Þ 2
A2 ¼ b2
3a2 c2 ; B2 ¼ b2 c2 9a2 e2 , C 2 ¼ c22 3b2 e2 ; D2 ¼ B22 4A2 C 2 ; h ¼ arccos qffiffiffiffiffiffi T 0 , and T 0 ¼ ð2A2 b2 3a2 B2 Þ= 2 A32 . Note that, different from Eqs. (8) and (20), since D2 < 0, the solution given by the S. Fan formulas has a different form as shown in Eq. (26). Hence, the intersection voltage becomes V gs;sub ¼ V go þ V off þ /n . As a result, the corresponding gate voltages at the intersection points in different regions have been determined. A unified Ef ;u expression suitable in all the operation regions can be obtained by using the smoothing function, i.e.,
Ef ;u ¼
Ef ;sub 1 þ exp½ðV gs V gs;sub Þ= þ
Ef ;mod 1 þ exp½ðV gs V gs;E0 Þ= þ exp½ðV gs V gs;sub Þ=
ð27Þ
Ef ;0 þ 1 þ exp½ðV gs V gs;E1 Þ= þ exp½ðV gs V gs;E0 Þ= þ
Ef ;1 : 1 þ exp½ðV gs V gs;E1 Þ=
To obtain a more accurate solution, a correction term x is introduced, and the complete solution of Ef can be expressed as
Ef ¼ Ef ;u þ xðy; y0 ; y00 Þ ¼ Ef ;u
y=y0 1 0:5yy00 =y0 =y0
y ¼ exp½bðV go Ef ;u Þ " !# Ef ;u c0 A2=3 ðV go Ef ;u Þ2=3 1 þ exp /th " !# Ef ;u c1 A2=3 ðV go Ef ;u Þ2=3 1 þ exp /th
ð28Þ
ð29Þ
where b ¼ A=D/th , and y0 and y00 are the first- and second-order derivatives of y, respectively. Furthermore, taking the bottom of the conduction band as a reference, it is easy to find the surface potential as ws ¼ Ef þ /n . 3. Drain current model
Ef ¼ E0 ) is calculated by [2] V gs;E0 ¼ D/Ath ln 2 þ c0 ðD/th ln 2Þ þ V off þ /n . Furthermore, the moderate 2DEG and subthreshold regions intersect at the gate voltage (V gs;sub ) which is extracted by the condition of Ef ¼ 0. Therefore, from Eq. (2), one obtains
"
V go ¼
b2 þ
ð22Þ
Moreover, the corresponding gate voltage (V gs;E0 ) at the intersection point of the strong and moderate 2DEG regions (i.e.,
E0 E1 E0 þ exp D/th exp : ns ¼ D/th exp /th /th /th
ð25Þ
where j ¼ /th =c0 and r ¼ A=ðDc0 Þ. Analogously, using the S. Fan formulas [9], the solution becomes
ð15Þ
where s ¼ D/th expðc0 k1 =/th Þ. Eq. (15) can be solved explicitly by using the Lambert W function (W 0 ) as
Ef ;mod ¼ V go
ðAV go Þ2 ¼ ðj rV go Þ3
ð24Þ
Similarly, Eq. (24) is also re-expressed as a cubic equation, i.e.,
Based on the definition of the surface potential, the drain current Ids0 can be written as [2]
eleff W ðV gs V off wm þ /th Þw Ids0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 L d 1þ# w
ð30Þ
where W and L are the gate width and length, respectively, # is a fitting parameter, w ¼ wsL ws0 ; wm ¼ ðwsL þ ws0 Þ=2, and ws0 and wsL can be calculated using Eq. (28) by replacing /n with the applied source and drain voltages, respectively. It is observed that, the carrier velocity has a nonmonotonic dependence on the electric field perpendicular to the channel. Therefore, a phenomenological model for the effective carrier mobility leff is introduced [2], in which the quantum well structure is not included, i.e.,
W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
leff ¼
l0 1 þ p1 ðV G0 ws Þ þ p2 ðV G0 ws Þ2
ð31Þ
where l0 is the low-field carrier mobility, p1 and p2 are the degradation parameters, and V G0 ¼ V gs V off . In addition, the model for self-heating effect in static case can be included as follows [5]:
ID ¼
b0 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b0 4a0 c0 2a0
ð32Þ
where a0 ¼ V ds Rth ; b0 ¼ T 1 bT0 Ids0 V ds Rth , and c0 ¼ Ids0 T. Herein, Rth is the thermal resistance of the device, T is room temperature, and b0 is a fitting parameter. The so-called current collapse manifests itself as a reduction of the current when a large alternating signal is applied. It is assumed that current collapse in the DC I–V characteristics is observed when slow voltage sweeps are adopted and quasi-steady-state can be used here. According to the results from measurements [10], the source and drain access resistances (Rds ) are responsible for the current collapse. An increase in Rds leads to a decrease of the device current [11]. As a result, the current collapse becomes significant in pulsed-IV characteristics. As shown in [12], the current collapse begins at V ds ¼ V knee . After the current collapse, there is a sudden increase in drain current at a certain value of drain-tosource voltage defined as V ds ¼ V kink . Then, the drain current recovers to its normal value in the post-kink region (V ds > V kink ). In other words, the current collapse occurs in the V knee < V ds < V kink region. From [11], a semi-empirical Rds model accounting for the current collapse is developed as
Rds ¼ R0 þ
f DR 1 a tanhðvÞ
57
ð33Þ
where R0 is the drain and source access resistances without considering current collapse, DR is the increase of resistances, and a is a fitting parameter close to unity. Parameter v is related to the electric field near the drain edge and modeled as v ¼ ðV ds V knee Þ=P knee . In this paper, R0 and DR are treated as fitting parameters. In addition, the function f restricts current collapse to V knee < V ds < V kink region and f ¼ f1 þ exp½2ðV ds V kink Þ þ exp½2ðV knee V ds Þg1 . Since a is very close to 1, Eq. (33) can be approximated as
Rds ¼ R0 þ
f DR 2v f DR 2v e þ 1 R0 þ e : 2 2
ð34Þ
As a result, the drain current of the unified model is given by
Ids ¼
ID ð1 þ kV dseff Þ 1 þ Rds g chi
ð35Þ
where g chi ¼ qWns leff =L; V dseff is the effective drain-to-source voltage, and k is a parameter accounting for the channel-length modulation. Compared to other Rds models which relies on empirical functions [7] or very complex expressions [11], our model is a trade-off between accuracy and efficiency. 4. Results and discussions The surface potential (or Ef ) and its derivatives in different operation regions under different sets of fc0 ; c1 ; Dg parameters are shown in Fig. 3, together with the numerical results as
Fig. 3. Comparison of Ef calculation between the explicit solution and the numerical results for different sets of parameters.
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W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
Fig. 4. Comparison of analytical model (lines) and experimental data (symbols) [13] for (a) drain current Ids and transconductance g m vs. V gs characteristics and (b) Ids vs. V ds characteristics. Parameters used for simulations: d ¼ 21 nm, L ¼ 1 lm, V off ¼ 3:12 V, # ¼ 0:1 V1 ; l0 ¼ 0:085 m2 /(Vs), p1 ¼ 3:4 109 V1 ; p2 ¼ 3:5 1017 V2 ; Rds ¼ 4 X; k ¼ 0 V1 ; Rth ¼ 1:2 K/W, and b0 ¼ 500 K.
comparisons. An excellent agreement is achieved between the explicit solution of Eq. (28) and the numerical results from the subthreshold to strong 2DEG regions. In Fig. 3(a) and (b), the boundary conditions for different regions are plotted, i.e., Ef ¼ E1 at V gs;E1 ; Ef ¼ E0 at V gs;E0 and Ef ¼ 0 at V gs;sub . It is clearly shown that for different sets of fc0 ; c1 ; Dg parameters, the values of boundary are different. In Fig. 3(b), the second subband E1 can be ignored, but in Fig. 3(a) and (c), the contribution of E1 should be included. Notice that, sets of fc0 ; c1 ; Dg parameters in Fig. 3(a) and (c) are for GaAs-based HEMTs and that in Fig. 3(b) is for GaN-based HEMTs. Therefore, the proposed model is valuable for different types of HEMTs. In particular, the partial derivatives of the surface potential with respect to the gate voltage are shown in Fig. 3(a), and it demonstrates the smoothness of our model. In addition, the regional asymptotic approximations only valid in their respective modeled regions are indicated in Fig. 3(c). With smoothing function as used in Eq. (27), these regional solutions can be unified. Furthermore, the preciseness of the complete calculation is depicted in Fig. 3(d), which shows that the maximum of the absolute errors occurs at the moderate 2DEG region and a high precision is obtained at the subthreshold and strong 2DEG regions. The overall errors with and without E1 are also compared, and the solutions considering E1 improve the accuracy. In the case of Fig. 3(b) which is for GaN-based HEMTs, the results with and without E1 are close especially in the strong 2DEG region. Nevertheless, in the moderate 2DEG and subthreshold regions, the model accounting for E1 gets better results. For the case in Fig. 3(c), there is a large discrepancy for the result without E1 . As shown in Fig. 3 (c) and (d), for certain types of devices, E1 is significant and cannot be ignored in normal operating regions. The explicit drain current model described in the previous sections has been verified on the AlGaN/GaN HEMTs [13] in Fig. 4. For the GaN-based HEMTs, the fc0 ; c1 ; Dg parameters can be set to the values in Fig. 3(b). Fig. 4(a) shows the drain current in logarithmic scale to verify the model prediction in the subthreshold and moderate 2DEG regions. Furthermore, the transconductance g m characteristics are accurately predicted by the new model, which are determined by the accurate descriptions of surface potential, mobility reduction, self heating effect and series resistances. In Fig. 4(b), the output characteristics are also demonstrated. Current collapse is suppressed due to the effective passivation technique [13]. Consequently, current collapse is insignificant in Fig. 4(b). In order to verify the description of current collapse, the behavior of drain current in AlGaN/GaN HEMTs is depicted in Fig. 5. When V ds varies up from 0 to 12 V, the presence of strong current
Fig. 5. Measured [12] (symbols) and modeled (lines) Ids vs. V ds characteristics. The drain bias has first been increased from 0 to 12 V (red markers) and then decreased from 12 to 0 V (blue markers). Parameters used for simulations: d ¼ 30 nm, W ¼ 100 lm, L ¼ 0:3 lm, V off ¼ 7 V, # ¼ 0:08 V1 ; l0 ¼ 0:03 m2 /(Vs), p1 ¼ 6109 V1 ; p2 ¼ 61017 V2 ; R0 ¼ 0:5 X; tknee ¼ 5 V, V gT ¼ 5 V, tkink ¼ 9:4 V, and g ¼ 0:857. In addition, for red solid lines DR ¼ 8 X and P knee ¼ 4 V, and for blue dashed lines DR ¼ 0 X. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
collapse is observed at low drain biases. Then, a sudden increase of Ids occurs, which is called kink effect. According to [12], hot electron trapped in the gate-drain space which increases the gatedrain series resistance results in a current collapse. In the postkink region (V ds > V kink ), Ids returns to its normal value. Therefore, parameters V knee and V kink determine the range of drain voltage where current collapse occurs at. As indicated in [12], V knee and V kink vary with gate voltages, and thus, the expressions of V knee and V kink have been extended as linear functions of V gs . For simplic1=4
ity, V knee ¼ tknee þ V gs =½1 þ ðV gs =V gT Þ4 and V kink ¼ gV gs þ tkink , where tknee ; V gT ; g, and tkink are fitting parameters. Moreover, no current collapse is detected if V ds is swept down from 12 to 0 V as shown in Fig. 5 (blue dashed line). Since parameters DR and Pknee determine how strong the current collapse affects the drain current, DR is set to 0 in this case. From the explanation in [14], this is because the traps are empty during a downward drain sweep and electrons are not hot carriers at the end of the sweep. It is clearly seen from Fig. 5, our model can capture these characteristics well. In addition, the self-heating effect is not included, and
W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
the behavior of upward or downward V ds pumping is similar to that of poststress or prestress conditions. 5. Conclusions Based on the two lowest subbands in the quantum triangular well, an explicit approximation for Ef and surface potential as a function of terminal voltages has been developed. This solution is valid for various types of HEMTs and results in an accurate description of all the operation regions. Furthermore, the proposed drain current model is based on surface potential formulations and describes the drain current using one single-piece equation. In addition, current collapse is captured by a semi-empirical model of the source and drain access resistances. The presented model shows a good agreement with the numerical and experimental results. It can be easily incorporated in device-circuit simulators. Acknowledgment This work was supported by Guangdong Natural Science Foundation (Grants Nos. S2013010013088 and 2014A030313366). References [1] Kola S, Golio JM, Maracas GN. An analytical expression for Fermi level versus sheet carrier concentration for HEMT modeling. IEEE Electron Device Lett 1988;9(3):136–8. [2] Cheng X, Wang Y. A surface-potential-based compact model for AlGaN/GaN MODFETs. IEEE Trans Electron Devices 2011;58(2):448–54.
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[3] Khandelwal S, Goyal N, Fjeldly TA. A physics-based analytical model for 2DEG charge density in AlGaN/GaN HEMT devices. IEEE Trans Electron Devices 2011;58(10):3622–5. [4] Khandelwal S, Fjeldly TA. A physics based compact model of I–V and C–V characteristics in AlGaN/GaN HEMT devices. Solid-State Electron 2012;76: 60–6. [5] Deng W, Huang J, Ma X, Liou JJ. An explicit surface potential calculation and compact current model for AlGaN/GaN HEMTs. IEEE Electron Device Lett 2015;36(2):108–10. [6] Khandelwal S, Chauhan YS, Fjeldly TA. Analytical modeling of surface-potential and intrinsic charges in AlGaN/GaN HEMT devices. IEEE Trans Electron Devices 2012;59(10):2856–60. [7] Zhang J, Syamal B, Zhou X, Arulkumaran S, Ng GI. A compact model for generic MIS-HEMTs based on the unified 2DEG density expression. IEEE Trans Electron Devices 2014;61(2):314–23. [8] Zhang A, Zhang L, Tang Z, Cheng X, Wang Y, Chen KJ, et al. Analytical modeling of capacitances for GaN HEMTs, including parasitic components. IEEE Trans Electron Devices 2014;61(3):755–61. [9] Fan S. A new extracting formula and a new distinguishing means on the one variable cubic equation. Nat Sci J Hainan Teach Coll 1989;2(2):91–8. [10] Simin G, Koudymov A, Tarakji A, Hu X, Yang J, Asif Khan M, et al. Induced strain mechanism of current collapse in AlGaN/GaN heterostructure field effect transistors. Appl Phys Lett 2001;79(16):2651–3. [11] Koudymov A, Shur MS, Simin G. Compact model of current collapse in heterostructure field-effect transistors. IEEE Electron Device Lett 2007;28 (5):332–5. [12] Saadaoui S, Salem MMB, Gassoumi M, Maaref H, Gaquiére C. Anomaly and defects characterization by I–V and current deep level transient spectroscopy of Al0.25Ga0.75N/GaN/SiC high electron-mobility transistors. J Appl Phys 2012;111:073713. [13] Tang Z, Huang S, Jiang Q, Liu S, Liu C, Chen KJ. High-voltage (600-V) lowleakage low-current-collapse AlGaN/GaN HEMTs with AlN/SiNx passivation. IEEE Electron Device Lett 2013;34(3):366–8. [14] Wang M, Chen KJ. Kink effect in AlGaN/GaN HEMTs induced by drain and gate pumping. IEEE Electron Device Lett 2011;32(4):482–4.
Contents lists available at ScienceDirect
Solid-State Electronics journal homepage: www.elsevier.com/locate/sse
A compact drain current model for heterostructure HEMTs including 2DEG density solution with two subbands Wanling Deng a,b,⇑, Junkai Huang a, Xiaoyu Ma a, Juin J. Liou b, Fei Yu a a b
Department of Electronic Engineering, Jinan University, Guangzhou 510630, China Department of Electronic Engineering, University of Central Florida, Orlando 32826, USA
a r t i c l e
i n f o
Article history: Received 23 June 2015 Received in revised form 24 September 2015 Accepted 16 October 2015 Available online 11 November 2015 Keywords: High electron mobility transistors (HEMTs) Surface potential Fermi level Drain current model
a b s t r a c t An explicit and precise model for two dimensional electron gas (2DEG) charge density and Fermi level (Ef ) in heterostructure high electron mobility transistors (HEMTs) is developed. This model is from a con€dinger’s and Poisson’s equations in the quantum well with two important energy sistent solution of Schro levels. With these closed-form solutions, a unified surface potential calculation valid for all the operation regions is derived. With the help of surface potential, a single-piece drain current model is developed which is also capable of describing the current collapse effect by using a semi-empirical expression of source/drain access region resistances. Comparisons with numerical and measured data show that the proposed model gives an accurate description of Ef and drain current in all regions of operation. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, heterostructure high electron mobility transistors (HEMTs) have attracted much attention in high-speed and high-power applications. One of the most interesting properties of these devices is the formation of the two dimensional electron gas (2DEG) with a very high electron mobility at the heterointerface. From the computer-aided design perspective, there is still an urgent demand for an accurate and computationally efficient compact DC model for HEMTs. The formation of 2DEG in the quantum well near the heterointerface is the main principle of the HEMT device operation, and the modeling of 2DEG sheet carrier density (ns ) is a basic requirement in the development of a compact model for these devices. The calculation of ns has to be obtained from the well-known charge control equations [1] as a self-consistent solution of Poisson’s and €dinger’s equations in the quantum well. This quantum well Schro can be approximated as a triangular well and has two lowest subbands (E0 and E1 ). The major challenge in calculating ns is due to the complicated transcendental equation of ns which varies with the gate biases (V gs ). Obviously, the transcendental equation is not suitable for compact modeling. Recently, the physical-based approximations [2–4] for explicit solution of ns were developed and the surface-potential-based drain current models were also ⇑ Corresponding author at: Department of Electronic Engineering, Jinan University, Guangzhou 510630, China. Tel.: +86 020 85222481; fax: +86 020 85220231. E-mail address: [email protected] (W. Deng). http://dx.doi.org/10.1016/j.sse.2015.10.005 0038-1101/Ó 2015 Elsevier Ltd. All rights reserved.
presented. However, their calculations were derived by only considering one subband (E0 ) in the quantum well and ignoring the other one (E1 ). In our previous work [5], we also used this assumption and presented an explicit solution to efficiently compute ns and the quasi-fermi potential Ef . Compared with the models in [2–4], our scheme [5] is more straightforward and accurate. Nevertheless, as indicated in [6,7], the assumption of neglecting the contribution of the second subband (E1 ) is only valid for certain types of devices, and E1 has a significant impact on device characteristics when device parameters are varied. In [6], an initial result without considering E1 was obtained first. To enhance the accuracy of the initial, refinements including the contribution from E1 were carried out by Householder’s method for solving implicit functions. In addition, [6] does not take the case of Ef > E1 into account which is important for GaAs-based HEMTs. Therefore, Zhang et al. [7] improved the model in [6], and analytical expressions for ns and Ef as explicit functions of the terminal biases were proposed. However, this model is very complex. Thus, it is very important to achieve a simply but accurate solution for compact models. In this paper, accounting for the two lowest subbands in the quantum well, we present an improved analytical calculation for ns and Ef based on our previous work [5]. After that, the surface potential (ws ) calculation can be obtained using Ef . Therefore, a surface-potential-based compact model for HEMTs is developed to predict the current–voltage (I–V) characteristics. Furthermore, the current collapse in I–V characteristics is also captured by using a semi-empirical model for source/drain access region resistances.
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2. The calculation of the Fermi level
where
b ¼ 3ðA=DÞV go ð2 þ A=DÞ2 ðc0 þ c1 Þ3 A2 ,
a ¼ ð2 þ A=DÞ3 ,
c ¼ 3ðA=DÞ V 2go ð2 þ A=DÞ þ 2ðc0 þ c1 Þ3 A2 V go , and e ¼ ðA=DÞ3 V 3go 2
€dinger’s As shown in Fig. 1, a self-consistent solution of Schro and Poisson’s equations in the triangular well including two subbands, is given by [7]
ns ¼
e qd
ðV gs V off /n Ef Þ
ð1Þ
ð2Þ
E0 ¼ c0 ns2=3
ð3Þ
E1 ¼ c1 ns2=3
ð4Þ
where e and d are the permittivity and thickness of the material between the gate and 2DEG, respectively, V off is the cutoff voltage, Ef is the Fermi potential with respect to the bottom of conduction band, /n is the channel potential, D is Density of States, c0 and c1 are determined by Robin boundary condition [8], and /th is the thermal voltage. Note that, Eqs. (1)–(4) are based on the one€dinger’s equations, but for dimensional (1D) Poisson’s and Schro high voltage and high field modes, the two-dimensional system has to be solved. For simplification, we only apply the 1D Poisson’s €dinger’s equations in this paper. and Schro As indicated by Zhang et al. [7], different sets of parameters fc0 ; c1 ; Dg lead to different contributions of E0 and E1 . Therefore, this variation results in a more complicated solution of Ef and ns . Obviously, as shown in Eqs. (1)–(4), the exact solutions of Ef and ns are transcendental in nature. To obtain a computationally efficient solution, which is suitable for circuit simulators, the regional approach [3] is employed here. As a result, three operation regions are divided, i.e., the strong 2DEG region, the moderate 2DEG region, and the subthreshold region.
1=3
b Y 1 Y 2 3a
ð8Þ
Y 1;2 ¼ A0 b þ 1:5aðB0
pffiffiffiffi DÞ
ð9Þ
where Ef ;1 denotes the Fermi potential in the case (A), D ¼ B20 2
4A0 C 0 ; A0 ¼ b 3ac; B0 ¼ bc 9ae, and C 0 ¼ c2 3be. For case (B), Eq. (2) is reduced to
ns ¼ DðEf E0 Þ ¼ DðEf c0 n2=3 s Þ:
ð10Þ
Eq. (10) is the same as the case where E1 is negligible, which can be solved in the same way as given before [5]. As a consequence, the solution for case (B) is marked as Ef ;0 . 2.2. Moderate 2DEG region In the moderate 2DEG region, we have 0 < Ef < E0 . Therefore, Eq. (2) can be simplified as [7]
! Ef c0 n2=3 s : /th
ns ¼ D/th exp
ð11Þ
It should be noted that, from [1], Eq. (2) can be re-expressed as
Ef ¼ /th lnðYÞ RþS þ Y¼ 2
ð12Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 RþS RSð1 ens =D/th Þ 2
between Ef and n2=3 using Eqs. (12) and (13) in Fig. 2, which shows s
Here, we assume that E1 > E0 . In the strong 2DEG region, we have Ef > E0 . Thus, there exist two possible conditions: (A) Ef > E1 and (B) E0 < Ef < E1 . For case (A), Eq. (2) can be approximated as
ð5Þ
Using Eq. (1), Ef as a function of V gs can be expressed as
2=3
a linear approximation between Ef and ns . Consequently, Eq. (11) can be approximated as
Ef c0 ðk3 Ef þ k1 Þ /th ð1 c0 k3 ÞEf c0 k1 ¼ D/th exp exp ; /th /th
ns ¼ D/th exp
ð6Þ
where A ¼ e=qd and V go ¼ V gs V off /n . Except for some coefficients, Eq. (6) is very similar to the relation of Ef vs. V gs in our previous work [5], but [5] neglected the contribution of E1 . Furthermore, Eq. (6) can be rewritten as a cubic equation 2
aE3f þ bEf þ cEf þ e ¼ 0
Fig. 1. Energy-band diagram of a HEMT.
ð13Þ
where R ¼ e1=/th E0 ; S ¼ e1=/th E1 , and Y ¼ e1=/th Ef . We plot the relation
2.1. Strong 2DEG region
AðV go Ef Þ ¼ 2DEf Dðc0 þ c1 ÞA2=3 ðV go Ef Þ2=3
1=3
Ef ;1 ¼
Ef E0 Ef E1 þ ln 1 þ exp ns ¼ D/th ln 1 þ exp /th /th
ns ¼ DðEf E0 Þ þ DðEf E1 Þ ¼ 2DEf Dðc0 þ c1 Þn2=3 s :
ðc0 þ c1 Þ3 A2 V 2go . The explicit solution of Ef can be calculated by the S. Fan formulas [9]
ð7Þ
Fig. 2. Ef as a function of n2=3 s .
ð14Þ
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W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
where k1 and k3 are fitting parameters, which can be extracted using two points in Fig. 2 to make a line. Substituting Eq. (1) into Eq. (14), we have
AðV go Ef Þ ¼ s exp
ð1 c0 k3 ÞEf /th
/th W0 1 c0 k3
sð1 c0 k3 Þ A/th
exp
V go /th =ð1 c0 k3 Þ
ð16Þ
where Ef ;mod means the value of Ef in the moderate 2DEG region. 2.3. Subthreshold region In the subthreshold region, Ef < 0, and therefore Ef E1 and Ef E0 become negative. The solution of the Fermi potential in this region (Ef ;sub ) is given by [7]
Ef ;sub ¼ V go /th W 0
2D V go : exp A /th
ð17Þ
2.4. Combining the regional models In the strong 2DEG region, the condition of Ef ¼ E1 determines the boundary between case (A) and case (B), at which the corresponding gate voltage is defined as V gs;E1 . When Ef ¼ E1 , Eq. (2) becomes
ns ¼ DðE1 E0 Þ þ D/th ln 2 ¼ Dðc1 c0 Þns2=3 þ D/th ln 2:
ð18Þ
Eq. (18) can be rewritten as a cubic equation, i.e.,
ðns MÞ3 ¼ n3 n2s
ð19Þ
where M ¼ D/th ln 2, and n ¼ Dðc1 c0 Þ. Herein, we still use the S. Fan formulas [9] to find an explicit solution
ns ¼
b1 Z 11=3 Z 1=3 2 3
Z 1;2 ¼ A1 b1 þ 1:5ðB1
ð20Þ pffiffiffiffi D1 Þ
ð21Þ 2
where b1 ¼ ð3M þ n3 Þ; c1 ¼ 3M2 ; e1 ¼ M3 , A1 ¼ b1 3c1 ; B1 ¼ B21
b1 c1 9e1 ; C 1 ¼ 3b1 e1 , and D1 ¼ 4A1 C 1 . As a consequence, from Eq. (1), V gs;E1 can be found as c21
V gs;E1 ¼
ns þ c1 ns2=3 þ V off þ /n : A
2=3
ð23Þ
A Taylor expansion to the first order is applied to further simplify Eq. (23). Combining with Eq. (2), yields
AV go ¼ D/th 1
c0 ðAV go Þ2=3 /th
#
:
pffiffiffi pffiffiffiffiffiffi A2 cos 3h þ 3 sin 3h 3a2
a2 ¼ r3 ; b2 ¼ A2 3jr2 ; c2 ¼ 3j2 r; e2 ¼ j3 ,
where
ð26Þ 2
A2 ¼ b2
3a2 c2 ; B2 ¼ b2 c2 9a2 e2 , C 2 ¼ c22 3b2 e2 ; D2 ¼ B22 4A2 C 2 ; h ¼ arccos qffiffiffiffiffiffi T 0 , and T 0 ¼ ð2A2 b2 3a2 B2 Þ= 2 A32 . Note that, different from Eqs. (8) and (20), since D2 < 0, the solution given by the S. Fan formulas has a different form as shown in Eq. (26). Hence, the intersection voltage becomes V gs;sub ¼ V go þ V off þ /n . As a result, the corresponding gate voltages at the intersection points in different regions have been determined. A unified Ef ;u expression suitable in all the operation regions can be obtained by using the smoothing function, i.e.,
Ef ;u ¼
Ef ;sub 1 þ exp½ðV gs V gs;sub Þ= þ
Ef ;mod 1 þ exp½ðV gs V gs;E0 Þ= þ exp½ðV gs V gs;sub Þ=
ð27Þ
Ef ;0 þ 1 þ exp½ðV gs V gs;E1 Þ= þ exp½ðV gs V gs;E0 Þ= þ
Ef ;1 : 1 þ exp½ðV gs V gs;E1 Þ=
To obtain a more accurate solution, a correction term x is introduced, and the complete solution of Ef can be expressed as
Ef ¼ Ef ;u þ xðy; y0 ; y00 Þ ¼ Ef ;u
y=y0 1 0:5yy00 =y0 =y0
y ¼ exp½bðV go Ef ;u Þ " !# Ef ;u c0 A2=3 ðV go Ef ;u Þ2=3 1 þ exp /th " !# Ef ;u c1 A2=3 ðV go Ef ;u Þ2=3 1 þ exp /th
ð28Þ
ð29Þ
where b ¼ A=D/th , and y0 and y00 are the first- and second-order derivatives of y, respectively. Furthermore, taking the bottom of the conduction band as a reference, it is easy to find the surface potential as ws ¼ Ef þ /n . 3. Drain current model
Ef ¼ E0 ) is calculated by [2] V gs;E0 ¼ D/Ath ln 2 þ c0 ðD/th ln 2Þ þ V off þ /n . Furthermore, the moderate 2DEG and subthreshold regions intersect at the gate voltage (V gs;sub ) which is extracted by the condition of Ef ¼ 0. Therefore, from Eq. (2), one obtains
"
V go ¼
b2 þ
ð22Þ
Moreover, the corresponding gate voltage (V gs;E0 ) at the intersection point of the strong and moderate 2DEG regions (i.e.,
E0 E1 E0 þ exp D/th exp : ns ¼ D/th exp /th /th /th
ð25Þ
where j ¼ /th =c0 and r ¼ A=ðDc0 Þ. Analogously, using the S. Fan formulas [9], the solution becomes
ð15Þ
where s ¼ D/th expðc0 k1 =/th Þ. Eq. (15) can be solved explicitly by using the Lambert W function (W 0 ) as
Ef ;mod ¼ V go
ðAV go Þ2 ¼ ðj rV go Þ3
ð24Þ
Similarly, Eq. (24) is also re-expressed as a cubic equation, i.e.,
Based on the definition of the surface potential, the drain current Ids0 can be written as [2]
eleff W ðV gs V off wm þ /th Þw Ids0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 L d 1þ# w
ð30Þ
where W and L are the gate width and length, respectively, # is a fitting parameter, w ¼ wsL ws0 ; wm ¼ ðwsL þ ws0 Þ=2, and ws0 and wsL can be calculated using Eq. (28) by replacing /n with the applied source and drain voltages, respectively. It is observed that, the carrier velocity has a nonmonotonic dependence on the electric field perpendicular to the channel. Therefore, a phenomenological model for the effective carrier mobility leff is introduced [2], in which the quantum well structure is not included, i.e.,
W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
leff ¼
l0 1 þ p1 ðV G0 ws Þ þ p2 ðV G0 ws Þ2
ð31Þ
where l0 is the low-field carrier mobility, p1 and p2 are the degradation parameters, and V G0 ¼ V gs V off . In addition, the model for self-heating effect in static case can be included as follows [5]:
ID ¼
b0 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b0 4a0 c0 2a0
ð32Þ
where a0 ¼ V ds Rth ; b0 ¼ T 1 bT0 Ids0 V ds Rth , and c0 ¼ Ids0 T. Herein, Rth is the thermal resistance of the device, T is room temperature, and b0 is a fitting parameter. The so-called current collapse manifests itself as a reduction of the current when a large alternating signal is applied. It is assumed that current collapse in the DC I–V characteristics is observed when slow voltage sweeps are adopted and quasi-steady-state can be used here. According to the results from measurements [10], the source and drain access resistances (Rds ) are responsible for the current collapse. An increase in Rds leads to a decrease of the device current [11]. As a result, the current collapse becomes significant in pulsed-IV characteristics. As shown in [12], the current collapse begins at V ds ¼ V knee . After the current collapse, there is a sudden increase in drain current at a certain value of drain-tosource voltage defined as V ds ¼ V kink . Then, the drain current recovers to its normal value in the post-kink region (V ds > V kink ). In other words, the current collapse occurs in the V knee < V ds < V kink region. From [11], a semi-empirical Rds model accounting for the current collapse is developed as
Rds ¼ R0 þ
f DR 1 a tanhðvÞ
57
ð33Þ
where R0 is the drain and source access resistances without considering current collapse, DR is the increase of resistances, and a is a fitting parameter close to unity. Parameter v is related to the electric field near the drain edge and modeled as v ¼ ðV ds V knee Þ=P knee . In this paper, R0 and DR are treated as fitting parameters. In addition, the function f restricts current collapse to V knee < V ds < V kink region and f ¼ f1 þ exp½2ðV ds V kink Þ þ exp½2ðV knee V ds Þg1 . Since a is very close to 1, Eq. (33) can be approximated as
Rds ¼ R0 þ
f DR 2v f DR 2v e þ 1 R0 þ e : 2 2
ð34Þ
As a result, the drain current of the unified model is given by
Ids ¼
ID ð1 þ kV dseff Þ 1 þ Rds g chi
ð35Þ
where g chi ¼ qWns leff =L; V dseff is the effective drain-to-source voltage, and k is a parameter accounting for the channel-length modulation. Compared to other Rds models which relies on empirical functions [7] or very complex expressions [11], our model is a trade-off between accuracy and efficiency. 4. Results and discussions The surface potential (or Ef ) and its derivatives in different operation regions under different sets of fc0 ; c1 ; Dg parameters are shown in Fig. 3, together with the numerical results as
Fig. 3. Comparison of Ef calculation between the explicit solution and the numerical results for different sets of parameters.
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W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
Fig. 4. Comparison of analytical model (lines) and experimental data (symbols) [13] for (a) drain current Ids and transconductance g m vs. V gs characteristics and (b) Ids vs. V ds characteristics. Parameters used for simulations: d ¼ 21 nm, L ¼ 1 lm, V off ¼ 3:12 V, # ¼ 0:1 V1 ; l0 ¼ 0:085 m2 /(Vs), p1 ¼ 3:4 109 V1 ; p2 ¼ 3:5 1017 V2 ; Rds ¼ 4 X; k ¼ 0 V1 ; Rth ¼ 1:2 K/W, and b0 ¼ 500 K.
comparisons. An excellent agreement is achieved between the explicit solution of Eq. (28) and the numerical results from the subthreshold to strong 2DEG regions. In Fig. 3(a) and (b), the boundary conditions for different regions are plotted, i.e., Ef ¼ E1 at V gs;E1 ; Ef ¼ E0 at V gs;E0 and Ef ¼ 0 at V gs;sub . It is clearly shown that for different sets of fc0 ; c1 ; Dg parameters, the values of boundary are different. In Fig. 3(b), the second subband E1 can be ignored, but in Fig. 3(a) and (c), the contribution of E1 should be included. Notice that, sets of fc0 ; c1 ; Dg parameters in Fig. 3(a) and (c) are for GaAs-based HEMTs and that in Fig. 3(b) is for GaN-based HEMTs. Therefore, the proposed model is valuable for different types of HEMTs. In particular, the partial derivatives of the surface potential with respect to the gate voltage are shown in Fig. 3(a), and it demonstrates the smoothness of our model. In addition, the regional asymptotic approximations only valid in their respective modeled regions are indicated in Fig. 3(c). With smoothing function as used in Eq. (27), these regional solutions can be unified. Furthermore, the preciseness of the complete calculation is depicted in Fig. 3(d), which shows that the maximum of the absolute errors occurs at the moderate 2DEG region and a high precision is obtained at the subthreshold and strong 2DEG regions. The overall errors with and without E1 are also compared, and the solutions considering E1 improve the accuracy. In the case of Fig. 3(b) which is for GaN-based HEMTs, the results with and without E1 are close especially in the strong 2DEG region. Nevertheless, in the moderate 2DEG and subthreshold regions, the model accounting for E1 gets better results. For the case in Fig. 3(c), there is a large discrepancy for the result without E1 . As shown in Fig. 3 (c) and (d), for certain types of devices, E1 is significant and cannot be ignored in normal operating regions. The explicit drain current model described in the previous sections has been verified on the AlGaN/GaN HEMTs [13] in Fig. 4. For the GaN-based HEMTs, the fc0 ; c1 ; Dg parameters can be set to the values in Fig. 3(b). Fig. 4(a) shows the drain current in logarithmic scale to verify the model prediction in the subthreshold and moderate 2DEG regions. Furthermore, the transconductance g m characteristics are accurately predicted by the new model, which are determined by the accurate descriptions of surface potential, mobility reduction, self heating effect and series resistances. In Fig. 4(b), the output characteristics are also demonstrated. Current collapse is suppressed due to the effective passivation technique [13]. Consequently, current collapse is insignificant in Fig. 4(b). In order to verify the description of current collapse, the behavior of drain current in AlGaN/GaN HEMTs is depicted in Fig. 5. When V ds varies up from 0 to 12 V, the presence of strong current
Fig. 5. Measured [12] (symbols) and modeled (lines) Ids vs. V ds characteristics. The drain bias has first been increased from 0 to 12 V (red markers) and then decreased from 12 to 0 V (blue markers). Parameters used for simulations: d ¼ 30 nm, W ¼ 100 lm, L ¼ 0:3 lm, V off ¼ 7 V, # ¼ 0:08 V1 ; l0 ¼ 0:03 m2 /(Vs), p1 ¼ 6109 V1 ; p2 ¼ 61017 V2 ; R0 ¼ 0:5 X; tknee ¼ 5 V, V gT ¼ 5 V, tkink ¼ 9:4 V, and g ¼ 0:857. In addition, for red solid lines DR ¼ 8 X and P knee ¼ 4 V, and for blue dashed lines DR ¼ 0 X. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
collapse is observed at low drain biases. Then, a sudden increase of Ids occurs, which is called kink effect. According to [12], hot electron trapped in the gate-drain space which increases the gatedrain series resistance results in a current collapse. In the postkink region (V ds > V kink ), Ids returns to its normal value. Therefore, parameters V knee and V kink determine the range of drain voltage where current collapse occurs at. As indicated in [12], V knee and V kink vary with gate voltages, and thus, the expressions of V knee and V kink have been extended as linear functions of V gs . For simplic1=4
ity, V knee ¼ tknee þ V gs =½1 þ ðV gs =V gT Þ4 and V kink ¼ gV gs þ tkink , where tknee ; V gT ; g, and tkink are fitting parameters. Moreover, no current collapse is detected if V ds is swept down from 12 to 0 V as shown in Fig. 5 (blue dashed line). Since parameters DR and Pknee determine how strong the current collapse affects the drain current, DR is set to 0 in this case. From the explanation in [14], this is because the traps are empty during a downward drain sweep and electrons are not hot carriers at the end of the sweep. It is clearly seen from Fig. 5, our model can capture these characteristics well. In addition, the self-heating effect is not included, and
W. Deng et al. / Solid-State Electronics 115 (2016) 54–59
the behavior of upward or downward V ds pumping is similar to that of poststress or prestress conditions. 5. Conclusions Based on the two lowest subbands in the quantum triangular well, an explicit approximation for Ef and surface potential as a function of terminal voltages has been developed. This solution is valid for various types of HEMTs and results in an accurate description of all the operation regions. Furthermore, the proposed drain current model is based on surface potential formulations and describes the drain current using one single-piece equation. In addition, current collapse is captured by a semi-empirical model of the source and drain access resistances. The presented model shows a good agreement with the numerical and experimental results. It can be easily incorporated in device-circuit simulators. Acknowledgment This work was supported by Guangdong Natural Science Foundation (Grants Nos. S2013010013088 and 2014A030313366). References [1] Kola S, Golio JM, Maracas GN. An analytical expression for Fermi level versus sheet carrier concentration for HEMT modeling. IEEE Electron Device Lett 1988;9(3):136–8. [2] Cheng X, Wang Y. A surface-potential-based compact model for AlGaN/GaN MODFETs. IEEE Trans Electron Devices 2011;58(2):448–54.
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