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A complete charge control model for HEMTs
Solid-State Electronics Vol. 41, No. I 1, pp. 1825-1826. 1997
Pergamon PII: S 0 0 3 8 - 1 1 0 1 ( 9 7 ) 0 0 1 4 1 - X
© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-1101/97 $17.00 + 0.00
NOTE A
COMPLETE CHARGE CONTROL MODEL FOR HEMTs (Received 6 February 1997; in revised f o r m 5 March 1997)
from the heterointerface[3,4] and Ed is the dielectric constant of the donor material. The threshold voltage is given by[2]
1. INTRODUCTION The modeling of H E M T s has been an area of active research since their fabrication[l]. The two-dimensional electron gas (2DEG) sheet density (ns) in a H E M T exhibits the following three regimes as the gate voltage (Vs) is varied-subthreshold, linear and saturation, n~ increases exponentially with Vs in the subthreshold regime where V~ is less than the threshold voltage, n, varies linearly with VB in the linear region and then gradually approaches a constant value in the saturation regime. The saturation regime is caused by the neutralization of the ionized donor atoms in the donor layer. The first H E M T model, by Delagebeaudeuf and Linh[2], assumed a linear relationship between ns and Vg. D r u m m o n d et al.[3] and Lee et al.[4] extended this model by considering the Fermi level variation with m. The linear models fail to predict the subthreshold current in the H E M T . Byun et al.[5] proposed the unified charge control model that correctly models the subthreshold and the linear regimes. However, all these models overestimate n~ in the saturation regime and hence fail to model the I - V curve there. Some authors have unified the linear and the saturation regime[6-8]. In this work, a complete charge control model for H E M T s is proposed. The model contains two fitting parameters that are extracted by comparing the model predictions with the results of self-consistent solution of the Schr6dinger equation and the Poisson equation. In Section 2, the model is described and compared with the computed n ~ - Vg data. The conclusions are presented in Section 3.
V~h = d?~,~- AE~ - qNdL:d
2Ed '
(4)
where Ores is the Schottky barrier height between the gate metal and the donor layer, AE~ is the conduction band offset at the heterointerface and Nd is the doping density in the donor layer. Equations (1), (3) and (4) describe the complete charge control model. We now compare this complete charge control model with results obtained from the self-consistent analysis. We plot the computed 2 D E G density (dots) and the 2DEG density obtained from the proposed model (solid lines) as a function of the gate voltage for three different H E M T structures in Fig. 1. The details of the H E M T structures and the values of the fitting parameters are given in Table 1. We see from Fig. 1 that the proposed charge control model is in excellent agreement with the computed data in the linear and the saturation regimes. To check the accuracy of the model in
1 0
....
i ....
i
-
i
.
.
.
.
i
8
/i
-
~
-
-
II
6
2. THE COMPLETE CHARGE CONTROL MODEL The n s - Vg relationship is given by V g - V,h = VTIn
+ V o ~ s
--
s
where V,h is the threshold voltage of the H E M T , Vr = kB T/q is the thermal voltage, no and n~o are two fitting parameters and V0 is a constant to be determined, n~ is the value at which ns saturates for large gate voltages. The first term on the right hand side describes the subthreshold regime while the second term describes both the linear and the saturation regimes. Above the subthreshold regime, we can neglect the first term. In the linear regime, we can also neglect (Vg - V,h) with respect to V0. Then eqn (1) reduces to ~ Vw - v , h ~ V~-V,h n, ~ n,0 (V~0+ (Vs -- Vt,)') '/' ~ n~0 V0
0 -1.5
i
-1
-0.5
0
0.5
.
i
1
.
.
.
.
1.5
G A T E V O L T A G E (V) Fig. 1. Comparison of the proposed model with the computed 2 D E G density in the linear and the saturation regimes for the three H E M T structures given in Tablc 1. The dots are the computed data and the lines are the model results.
(2)
To match this result with the linear model[2], we choose Vo = q ( L d + L s + Aa')nso
a2
(1)
(3)
Ed
where Ld is the donor layer thickness, L, is the spacer layer thickness, Ad = 80 A is the average separation of the 2 D E G
Table 1. The HEMT structures and the fitting parameters used for comparing the proposed model with computed data Nd (cm -~) Lo (/It) L, (/~) n,0 (era -2) no (cm-") I 1 x l0 n 200 50 .01 x 10~2 3.5 x 10" I1 2 x 10~s 200 100 .08 x 10" 00.2 x 10" III 1 x l0 is 500 200 4.8 x 10tl 0.15 x 10tt
1825
Note
1826 1012
3. CONCLUSION In conclusion, a complete charge control model was developed for HEMTs for the first time. The model describes very accurately the 2DEG density in the HEMT as a function of the gate voltage in the subthreshold, the linear and the saturation regimes. This model can be used to calculate the I - V characteristics.
1011
~
1010
Department of Electrical Engineering Princeton University Princeton, NJ 08544 U.S.A.
z 109
AMLANMAJUMDAR
REFERENCES
108
107 -1.5
-1
-0.5
0
0.5
1
1.5
GATE VOLTAGE (V) Fig. 2. Comparison of the proposed model with the computed 2DEG density in the subthreshold regime for the three HEMT structures given in Table 1. The dots are the computed data and the lines are the model results.
the subthreshold regime, we plot the 2DEG density on a logarithmic scale in Fig. 2. It is evident from Fig. 2 that this model is very accurate in the subthreshold regime too.
1. Mimura, T., Hiyamizu, S., Josben, K. and Hikosaka, K., Jpn. J. Appl. Phys. 1981, 20, L317 2. Delagebeaudeuf, D. and Linh, N. T., IEEE Trans, Electron Devices, 1982, ED-29, 955 3. Drummond, T. J., Morkoc, H., Lee, K. and Shur, M., IEEE Electron Device Lett., 1982, EDL-3, 338 4. Lee, K., Shur, M. S., Drummond, T. J. and Morko¢, H., IEEE Trans. Electron Devices, 1983, ED-30, 207 5. Byun, Y. H., Lee, K. and Shur, M., IEEE Electron Device Lett., 1990, EDL-I1, 50 6. Rohdin, H. and Roblin, P., IEEE Trans. Electron Devices, 1996, ED-33, 664 7. Ahn, H. and El Nokali, M., IEEE Trans. Electron Devices, 1994, ED-41, 874 8. Guan, L., Christou, A., Halkias, G. and Barbe, D. F., 1EEE Trans. Electron Devices, 1995, ED-42, 612.
Pergamon PII: S 0 0 3 8 - 1 1 0 1 ( 9 7 ) 0 0 1 4 1 - X
© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-1101/97 $17.00 + 0.00
NOTE A
COMPLETE CHARGE CONTROL MODEL FOR HEMTs (Received 6 February 1997; in revised f o r m 5 March 1997)
from the heterointerface[3,4] and Ed is the dielectric constant of the donor material. The threshold voltage is given by[2]
1. INTRODUCTION The modeling of H E M T s has been an area of active research since their fabrication[l]. The two-dimensional electron gas (2DEG) sheet density (ns) in a H E M T exhibits the following three regimes as the gate voltage (Vs) is varied-subthreshold, linear and saturation, n~ increases exponentially with Vs in the subthreshold regime where V~ is less than the threshold voltage, n, varies linearly with VB in the linear region and then gradually approaches a constant value in the saturation regime. The saturation regime is caused by the neutralization of the ionized donor atoms in the donor layer. The first H E M T model, by Delagebeaudeuf and Linh[2], assumed a linear relationship between ns and Vg. D r u m m o n d et al.[3] and Lee et al.[4] extended this model by considering the Fermi level variation with m. The linear models fail to predict the subthreshold current in the H E M T . Byun et al.[5] proposed the unified charge control model that correctly models the subthreshold and the linear regimes. However, all these models overestimate n~ in the saturation regime and hence fail to model the I - V curve there. Some authors have unified the linear and the saturation regime[6-8]. In this work, a complete charge control model for H E M T s is proposed. The model contains two fitting parameters that are extracted by comparing the model predictions with the results of self-consistent solution of the Schr6dinger equation and the Poisson equation. In Section 2, the model is described and compared with the computed n ~ - Vg data. The conclusions are presented in Section 3.
V~h = d?~,~- AE~ - qNdL:d
2Ed '
(4)
where Ores is the Schottky barrier height between the gate metal and the donor layer, AE~ is the conduction band offset at the heterointerface and Nd is the doping density in the donor layer. Equations (1), (3) and (4) describe the complete charge control model. We now compare this complete charge control model with results obtained from the self-consistent analysis. We plot the computed 2 D E G density (dots) and the 2DEG density obtained from the proposed model (solid lines) as a function of the gate voltage for three different H E M T structures in Fig. 1. The details of the H E M T structures and the values of the fitting parameters are given in Table 1. We see from Fig. 1 that the proposed charge control model is in excellent agreement with the computed data in the linear and the saturation regimes. To check the accuracy of the model in
1 0
....
i ....
i
-
i
.
.
.
.
i
8
/i
-
~
-
-
II
6
2. THE COMPLETE CHARGE CONTROL MODEL The n s - Vg relationship is given by V g - V,h = VTIn
+ V o ~ s
--
s
where V,h is the threshold voltage of the H E M T , Vr = kB T/q is the thermal voltage, no and n~o are two fitting parameters and V0 is a constant to be determined, n~ is the value at which ns saturates for large gate voltages. The first term on the right hand side describes the subthreshold regime while the second term describes both the linear and the saturation regimes. Above the subthreshold regime, we can neglect the first term. In the linear regime, we can also neglect (Vg - V,h) with respect to V0. Then eqn (1) reduces to ~ Vw - v , h ~ V~-V,h n, ~ n,0 (V~0+ (Vs -- Vt,)') '/' ~ n~0 V0
0 -1.5
i
-1
-0.5
0
0.5
.
i
1
.
.
.
.
1.5
G A T E V O L T A G E (V) Fig. 1. Comparison of the proposed model with the computed 2 D E G density in the linear and the saturation regimes for the three H E M T structures given in Tablc 1. The dots are the computed data and the lines are the model results.
(2)
To match this result with the linear model[2], we choose Vo = q ( L d + L s + Aa')nso
a2
(1)
(3)
Ed
where Ld is the donor layer thickness, L, is the spacer layer thickness, Ad = 80 A is the average separation of the 2 D E G
Table 1. The HEMT structures and the fitting parameters used for comparing the proposed model with computed data Nd (cm -~) Lo (/It) L, (/~) n,0 (era -2) no (cm-") I 1 x l0 n 200 50 .01 x 10~2 3.5 x 10" I1 2 x 10~s 200 100 .08 x 10" 00.2 x 10" III 1 x l0 is 500 200 4.8 x 10tl 0.15 x 10tt
1825
Note
1826 1012
3. CONCLUSION In conclusion, a complete charge control model was developed for HEMTs for the first time. The model describes very accurately the 2DEG density in the HEMT as a function of the gate voltage in the subthreshold, the linear and the saturation regimes. This model can be used to calculate the I - V characteristics.
1011
~
1010
Department of Electrical Engineering Princeton University Princeton, NJ 08544 U.S.A.
z 109
AMLANMAJUMDAR
REFERENCES
108
107 -1.5
-1
-0.5
0
0.5
1
1.5
GATE VOLTAGE (V) Fig. 2. Comparison of the proposed model with the computed 2DEG density in the subthreshold regime for the three HEMT structures given in Table 1. The dots are the computed data and the lines are the model results.
the subthreshold regime, we plot the 2DEG density on a logarithmic scale in Fig. 2. It is evident from Fig. 2 that this model is very accurate in the subthreshold regime too.
1. Mimura, T., Hiyamizu, S., Josben, K. and Hikosaka, K., Jpn. J. Appl. Phys. 1981, 20, L317 2. Delagebeaudeuf, D. and Linh, N. T., IEEE Trans, Electron Devices, 1982, ED-29, 955 3. Drummond, T. J., Morkoc, H., Lee, K. and Shur, M., IEEE Electron Device Lett., 1982, EDL-3, 338 4. Lee, K., Shur, M. S., Drummond, T. J. and Morko¢, H., IEEE Trans. Electron Devices, 1983, ED-30, 207 5. Byun, Y. H., Lee, K. and Shur, M., IEEE Electron Device Lett., 1990, EDL-I1, 50 6. Rohdin, H. and Roblin, P., IEEE Trans. Electron Devices, 1996, ED-33, 664 7. Ahn, H. and El Nokali, M., IEEE Trans. Electron Devices, 1994, ED-41, 874 8. Guan, L., Christou, A., Halkias, G. and Barbe, D. F., 1EEE Trans. Electron Devices, 1995, ED-42, 612.