Electric Field Distribution Around Drain-Side Gate Edge ... - IEEE Xplore

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 10, OCTOBER 2013

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Electric Field Distribution Around Drain-Side Gate Edge in AlGaN/GaN HEMTs: Analytical Approach Jia Si, Jin Wei, Wanjun Chen, and Bo Zhang, Member, IEEE

Abstract— An analytical model is proposed in this paper for the surface electric field around the drain-side gate edge in the AlGaN/GaN HEMT to which the gate leakage, current collapse, and so on are highly related. Conformal mapping is implemented to mirror the device structure into a simplified geometry and the solution to the Laplace equation is thus achieved. Obtained from the proposed model, the surface electric field in the AlGaN/GaN HEMT shows a high peak around the drain-side gate edge, which is the cause for many reliability issues in HEMTs. The proposed model is then used to study the main factors that may impact the field distribution, such as drain voltage, thickness of barrier, and so on. Numerical simulations are carried out and compared with the analytical model. The high agreement between the results verifies the proposed model. Index Terms— Analytical model, gate-edge electric field, HEMTs.

I. I NTRODUCTION

A

LGaN/GaN HEMTs are promising as the next generation high frequency, high power devices for the wide bandgap energy, high electron mobility, capability to operate at high temperature [1], [2], and so on. Several issues are, however, strongly limiting their performance and reliability, such as the gate leakage, OFF-state breakdown, current collapse [3]–[6], and so on. The electric field distribution, especially the surface field around the drain-side gate edge in the AlGaN/GaN HEMT, is one of the most important factors related with these issues. Therefore, understanding the surface electric field distribution is worthwhile for designing high-performance AlGaN/GaN HEMTs. In silicon high-power devices such as Vertical Double-diffused MOSFET and Insulated Gate Bipolar Transistor, the analysis of the electric field is well established [7], [8], but in AlGaN/GaN HEMT, there is still no analytical method to predict the electric field. Conformal mapping has been exploited previously to study silicon and GaAs devices [9]–[11]. Using this technique, Frensley [12] deduced the electric field distribution around drain-side gate edge in GaAs MESFET. The results, however, could not be transferred to AlGaN/GaN HEMT directly due to the different charge distribution. What is more, that work

Manuscript received March 30, 2013; revised May 26, 2013; accepted June 26, 2013. Date of publication July 19, 2013; date of current version September 18, 2013. The review of this paper was arranged by Editor G. Ghione. The authors are with the State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China ([email protected]; [email protected]; [email protected]; zhangbo@uestc. edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TED.2013.2272055

Fig. 1. (a) Charge distribution in the AlGaN/GaN HEMT at OFF state. (b) Setup of the physical model.

considered only the portion of field distribution between the gate and the drain, and not that under gate electrode that is strongly related to gate leakage and OFF-state breakdown. To fill this gap, an analytical model is proposed in this paper. Conformal mapping is implemented to transform the device structure into a new geometry to simplify the boundary conditions of the Laplace equation. The surface electric distribution around the drain-side gate edge is presented by the analytical model, and the main factors that influence the surface field distribution are studied. Numerical simulation has been carried out and verified the proposed model. II. A NALYTICAL M ODEL A. Mathematical Setup Fig. 1(a) shows the distribution of charges in an OFF-state GaN HEMT [13]. The polarization charges at the top surface of AlGaN barrier layer and the AlGaN/GaN interface induce a great polarization electric field [14]. In the undepleted regions, the 2-D electron gas (2-DEG) is provided by the ionization of surface states [15]. Thus, the positive charges on surface and electrons in channel largely offset the polarization electric field. The residual polarization electric field is small compared with the electric field around drain-side gate edge and could

0018-9383 © 2013 IEEE

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Fig. 2.


Boundary conditions.

be ignored, which would be verified later. Therefore, we could equal the density of 2-DEG and the polarization charges at the interface, and also the surface states charges and the surface polarization charges. In the depleted regions, the 2-DEG no longer exists so only polarization charges are located in the interface. The charging and discharging of surface states are ignored for simplicity (future work is needed to take them into consideration if more accuracy is required), and thus there are no net surface charges. Then, the electric field starts from positive polarization charges at the interface and ends at gate electrode. The mathematical setup of the proposed model is shown in Fig. 1(b). The electric field in GaN buffer is much lower than that in AlGaN barrier, so the influence of GaN buffer layer is simply ignored. The thickness of AlGaN barrier is a. A coordinate is built in which the region with x < 0 represents the undepleted region and the region with x > 0 represents the depleted region. The distance from the edge of the depletion region to gate edge is b, which is a function of the drain voltage and will be deduced later. The surface of AlGaN barrier is at y = 0, and the AlGaN/GaN interface at y = a. The charges at the source end have negligible influence on the distribution of electric field, so for simplicity, a semiinfinite region is used to represent the gate-to-source region without introducing a significant inaccuracy. Usually the AlGaN layer is unintentionally doped, so the doping concentration could be neglected and the potential could be obtained by solving the Laplace equation in the depletion region ∇ 2 V = 0.

(1)

Assuming the density of 2-DEG is σs (which is to be calculated according to [14]), then the boundary condition at y = a is as follows: ∂ V σs y=a = − ∂y εs . At the surface of the HEMT, the net charge concentration is regarded zero as explained earlier. Therefore, the corresponding boundary condition is as follows: ∂ V xb,y=0 = 0. The boundary conditions are shown in Fig. 2. It would be, however, very difficult to solve the Laplace equation as it is. Next, the geometry will be transformed by conformal mapping to make it easier [12], [16]. Detailed mapping steps are shown in Fig. 3. The original geometry is defined as z-plane in Fig. 3(a), where the points A, B, and C are at (b, 0), (0, 0), and (0, a), respectively. First, a hyperbolic cosine transformerz 1 = cosh(π z/a) maps the z-plane into z 1 -plane as shown in Fig. 3(b), where the points A,B, and C are at (cosh(πb/a), 0), (1, 0), and (−−, 0), respectively. Second, z 2 = (2z 1 − c+ 1)/(c+ 1) serves as a linear transformer to map point A to (1, 0), while point C remains at (−1, 0), as shown in Fig. 3(c), where c = cosh(πb/a). Finally, we apply an inverse hyperbolic transformation w(x, y) = π −1 cosh−1 (z 2 ) to unfold the geometry, as shown in Fig. 3(d). Thus in w-plane, A, B, and C are at (0, 0), (0, π −1 cosh−1 ((3 −c)/(1 +c))), and (0, 1), respectively. The total conformal transformation from z-plane to w-plane could be expressed as follows (2): 1 2 cosh(π z/a) − c + 1 cosh−1 . (2) π c+1 The boundary conditions have also been changed during the transformation. So, it is necessary to make corresponding revisions. w(x , y ) = x + i y =

SI et al.: ELECTRIC FIELD DISTRIBUTION AROUND DRAIN-SIDE GATE EDGE IN ALGaN/GaN HEMTs

The gate voltage remains zero, and the corresponding boundary condition is as follows: V y =0 = 0.

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where R is the distance between (x, 1) and M

R = (x − x 0 )2 + (1 − y0 )2 .

Therefore, the total potential caused by charges along y = 1 is obtained as follows (6): 0 σs (−x ) ln R d x (6) Vy =1 (M) = −∞ 2πεs +∞ σs (x ) ln R d x . + 2πεs 0

For the condition at x = 0, we have the following: ∂V ∂ V ∂y |x =0 = AB : y=0 × y=0 = 0 ∂x ∂y ∂x ∂V ∂x ∂V |x =0 = |x=0 × |x=0 = 0. BC : ∂x ∂x ∂x Therefore, the revised boundary condition is Similarly, the total potential caused by charges along ∂V |x =0 = 0. y = −1 is ∂x 0 The interface charges are redistributed in the transformed −σs (−x ) =−1 (M) = ln R d x (7) V y coordinate, and the revised boundary is expressed by the 2πεs −∞ following: +∞ σ (x ) s + ln R d x ∂V ∂y ∂V 2πεs 0 = × . y=a y =1 y =1 ∂y ∂y ∂y We can rewrite (2) as (3), where

(c + 1) cosh[π(x + i y )] + c − 1 a R = (x − x 0 )2 + (1 + y0 )2 −x . cosh−1 y = −i . π 2 The actual potential at point M(x 0 , y0 ) is the sum of the (3) (6) and (7). Further differentiation gives (8) V (M) = Vy =1 (M) + Vy =−1 (M) d y a(c + 1) sinh(π x ) +∞ . − x )2 + (1 + y )2 y =1 = 2 1/2 (x σ (x ) dy {[(c + 1) cosh(π x ) − (c − 1)] − 4} s 0 0 ln = )2 + (1 − y )2 d x . 2πε (x − x s 0 Therefore, the revised boundary condition is 0 0 ∂ V σs y =1 = − ∂y εs

As the voltage at B is the drain bias VD , so by substituting the coordinates of B(0, π −1 cosh−1 ((3 −c) /(1 +c))) into (8), the parameter c could be obtained as a function of VD . The parameter c is, however, in the integral part and is hard to extract directly. In this paper, the relation between VD and the parameter c is shown in Fig. 6 by calculating (8) at different c, and the corresponding c at a certain V D is in turn read from the figure.

where σs (x ) =

aσs (c + 1) sinh(π x )

1/2 [(c + 1) cosh(π x ) − (c − 1)]2 − 4 .

The revised boundary conditions are shown in Fig. 4. B. Potential in the Depletion Region Image charge method is used to solve the Laplace equation [17], [18]. To satisfy AC boundary in Fig. 4, image charge σs (−x) along y = 1 for x < 0 is introduced. This image charge will, however, destroy the boundary condition along the gate/barrier interface. Thus, addition image charges −σs (x) and −σs (−x) along y = −1 are introduced to fix it. The distribution of image charges is shown in Fig. 5. As the boundary conditions for the Laplace equation are conserved, the new charge distribution results in the same solution as that in Fig. 4. Now, let us begin to calculate the electric field and potential caused by the charge distribution. Taking the charges in xσs (x) at (x, 1) for x > 0, the electric field generated on the point M(x 0 , y0 ) is that in (4) while the corresponding potential at M(x 0 , y0 ) is shown in (5), −x σs (x ) 2π Rε +∞ s x σs (x ) Vx (M) = − Ed R = − ln R 2πεs R

E x (M) =

(4) (5)

C. Field Around Drain-Side Gate Edge—Portion in the Access Region As we know, the drain-side gate edge is usually where high electric field takes place. To calculate the value of electric field around the drain-side gate edge, we give the first-order Taylor expansion of potentialV near the point A in w-plane, as shown in (9). Because the gate metal lies at y = 0 and the electric field is vertical to the metal at its surface, the field along the x 0 direction is zero ∂V ∂V (9) V (x 0 , y0 ) = V |w=0 + w=0 x 0 + w=0 y0 ∂ x0 ∂y0 ∂ V = w=0 y0 ∂y0 From (8), we obtain the following: ∞ ∂V σs (x ) 2 | = dx w=0 ∂y0 πεs 0 1 + x 2 .

(10)

Let us consider the portion of surface field distribution in the access region between the drain and the gate, i.e., the region

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Fig. 4.


Boundary conditions for transformed coordinates.

Fig. 6. Relation between VD and c. When VD is determined, the corresponding c could be read from this figure.

the gate edge. dV d y0 dV = lim y0 →0 d y0

E(Y ) = lim

y0 →0

Fig. 5.

Distribution of mirror charges.

with 0 < x 0 < b. To calculate the electric field in z-plane, we consider the point X(x 0 , 0) in z-plane. Here, x 0 = b − ξ and ξ is a small positive number representing the distance to the gate edge. The corresponding y0 is obtained when point X (x 0 , 0) is mapped into w-plane y0

2 = I m(w) = π

c−1 c+1

1/4

πξ a

1/2 .

(11)

Thus, we get the expression of potential at X (x 0 , 0) 4σs c − 1 1/4 ξ 1/2 ∞ σs (x ) V (x 0 ) = 3/2 ( ) ( ) d x . (12) 2 ) π εs c + 1 a σ (1 + x s 0 The electric field is given by d V (x 0 ) d V (x 0 ) (13) = d x0 dξ 2σs c − 1 1/4 ∞ σs (x ) = 3/2 √ d x . σs (1 + x 2 ) π εs aξ c + 1 0

E(x 0 ) = −

D. Field Around Drain-Side Gate Edge—Portion Under the Gate Electrode The above procedure could not work in solving the portion of field distribution under the gate, i.e., the region with x 0 > b. The electric field under the gate electrode is perpendicular to the metal. Let us consider the point Y (b − ξ , y0 ) in z-plane, where y0 is very small and ξ is a small negative number whose absolute value represents the distance between Y and

(b−ξ,y

0)

(14)

d y (x0 ,y0 ) ÷ (x0 ,y0 ) . d y0

From (8), we could obtain the following: ∞ dV σs (x ) 2 lim = d x . πεs 0 1 + (x − x 0 )2 y0 →0 d y0 From (3), we could obtain the following: a(c + 1) sinh(π x 0 ) d y0 . (15) =

2 y0 →0 d y0 (c + 1) cosh(π x 0 ) + c − 1 − 4 lim

Thus, the electric field under the gate electrode could be expressed in w-plane as follows (16): ∞ σs (x ) 2 E(x 0 ) = dx (16) aπεs 0 1 + (x − x 0 )2

2 (c + 1) cosh(π x 0 ) + c − 1 − 4 . × (c + 1) sinh(π x 0 ) To obtain the electric field of the point Y (b − ξ , y0 ), we simply substitute the corresponding (x 0 , 0) into (16), where 2 c − 1 1/4 πξ 1/2 x 0 = Re[w(b − ξ )] = . (17) − π c+1 a III. R ESULTS AND D ISCUSSION Numerical simulations by Sentaurus TCAD are carried out to verify the proposed model. The parameters in the simulation are shown in Table I. In all the results, the point with ξ = 0 represents the drain-side gate edge with the drain at the positive end and source at the negative end, as defined in the inset of Fig. 7. In the simulated results, the field distribution with a very small distance from the surface is used to represent the surface


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TABLE I PARAMETERS OF D EVICE

Fig. 9. OFF-state surface electric field distributions at different drain voltages by the model without revision, the model with revision, and numerical simulation. VG is −7 V, the thickness of AlGaN layer is 24 nm, and Al fraction is 0.25 in the simulation.

Fig. 7. Simulated OFF-state electric field distribution at different depths from the surface. VG is −7 V, V D is 500 V, and the thickness of AlGaN layer a = 24 nm, Al fraction is 0.25.

are chosen in the rest of this paper for comparison with the analytical results. We have assumed the electric field in GaN layer is small enough to be ignored. Fig. 7 verified this assumption. The field at distances of >25 nm is much smaller than that of the surface field. According to (13) and (16), the electric field approaches infinity as ξ approaches zero, which could only happen in an ideal situation with an absolutely sharp gate corner. Therefore, the model could not achieve the field with a distance from the gate edge that approaches zero. For this reason, analytical field at ξ = +2 nm is used to represent the peak value of electric field. A. Electric Field Upon Drain Voltage

Fig. 8. Simulated and analytical OFF-state surface electric field distribution at different drain voltage V D of 0, 20, 200, 500, and 900 V, VG is −7 V, the thickness of AlGaN layer is 24 nm and Al fraction is 0.25.

field. Fig. 7 shows the simulated OFF-state field distributions around the drain-side gate edge with different distances to the surface. In this case, the gate voltage VG is −7 V, the drain voltage VD is 500 V, and the thickness of AlGaN barrier is 24 nm. Within several nanometers, the distributions are almost the same. When the distance is large, such as 15 nm, the distribution cannot represent the surface field. Therefore, the simulated distributions with a distance of 2 nm to the surface

The OFF-state surface electric field distributions around the drain-side gate edge in a AlGaN/GaN HEMT under different drain voltages (VD ) are shown in Fig. 8. The thickness of AlGaN layer is 24 nm. The gate is biased at − 7V in simulation to turn off the device. Generally, the analytical and the simulated results match very well, especially at relatively large drain voltages. As GaN HEMTs are aimed for high power application, the surface electric field at relatively large drain voltage is the main concern. Therefore, the model proves to be useful and accurate. At relatively small drain voltages, discrepancies occur between the simulation and the model. One reason may be the influence of unbalanced polarization field, but this influence is relatively very small by investigating the analytical and simulated field distribution at VD = 0 V. The region with ξ > 0 represents the gate-to-drain region, and the simulated field at this region should mainly be the unbalanced polarization field. As shown from the figure, the unbalanced polarization field is not a big influence. The influence upon the field distribution exerted by the gate voltage is the main cause for the discrepancies at small drain voltages. In our model, the gate voltage is treated as zero for simplicity, but an ordinary AlGaN/GaN HEMT need a

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Fig. 10. Simulated and analytical OFF-state surface electric field distribution at different barrier thicknesses of 16, 22, and 28 nm under VG = −4.5, −6, and −8.7 V, respectively. V D = 500 V, and Al fraction is 0.25.

negative gate bias to turn it off as the case in our simulation. To verify this, a small revision is made in the model. The gate voltage is counted in the drain voltage in the revision. For example, in the OFF-state condition with a VD of 20 V and a VG of −7 V, the revised model treats gate voltage as zero and drain voltage as 27 V. Fig. 9 shows the results of the revision at different drain voltages. At small drain voltages, the discrepancies between the model and the simulation are reduced by the revision, which could prove the influence of gate voltage in our model. At large drain voltages, the revision is not necessary.


Fig. 11. Simulated and analytical OFF-state surface electric field distributions at Al fractions of 0.22, 0.26, and 0.3 with VG = −5.3, −6.2, and −7.3 V, respectively. V D is set to be 500 V, and the thickness of AlGaN layer is 24 nm.

IV. C ONCLUSION An analytical model is proposed in this paper to extract the surface electric field distribution around the drain-side gate edge in AlGaN/GaN HEMTs. Techniques such as conformal mapping, mirror charges, and so on are exploited to solve the Laplace equation. Results from the analytical model agree well with numerical simulation. Based on the model, the drain voltage, the thicknesses, and Al fraction of AlGaN barrier have been studied and their influences on the field distribution are well analyzed. This model may provide an effective tool to understand many reliability issues related to the high electric field around the drain-side gate edge.

B. Electric Field Upon the Thickness of AlGaN Barrier We have also exploited our model to study the influence of thickness of AlGaN barrier upon the surface electric field of GaN HEMTs. The thickness of AlGaN barrier may exert its influence on surface electric field through two aspects, the thickness itself and the change of 2-DEG density. Fig. 10 shows the surface electric field in AlGaN/GaN HEMTs with barrier thicknesses of 16, 22, and 28 nm at VD = 500 V in the simulation. As the threshold voltage changes with the barrier thickness, VG = −4.5, −6, and −8.7 V are used, respectively, to turn off the device in the simulation. Al fraction in AlGaN barrier is 0.25. The thicker the AlGaN barrier is, the higher the electric field is. The influence is, however, not very strong.

C. Electric Field Upon Al Fractions The 2-DEG density is a function of Al fraction in AlGaN barrier. Fig. 11 shows the surface electric field distributions in GaN HEMTs with different Al fraction of 0.22, 0.26, and 0.3 in their AlGaN barrier, respectively. In the simulation, gate voltages of −5.3, −6.2, and −7.3 V are used, respectively, to turn off the device. VD is set to be 500 V, and the barrier thickness is 24 nm. Apparently, higher Al fraction results in higher electric field.

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Jin Wei is currently pursuing the Ph.D. degree with the Hong Kong University of Science and Technology, Hong Kong. His current research interests include GaN devices, silicon power devices, and device modeling and simulation.

Jia Si is currently pursuing the M.S. degree in nanoelectronic devices and integrated circuits with Peking University, Beijing, China. Her current research interests include graphene and CNT transistor, GaN device, and device modeling.

Bo Zhang (M’03) received the M.S. degree in electronic engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1988. He is currently a Full Professor with the University of Electronic Science and Technology of China.

Wanjun Chen received the Ph.D. degree in microelectronics and solid-state electronics from the University of Electronic Science and Technology of China, Chengdu, China, in 2007. He is currently an Associate Professor at UESTC.