586 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4. A P W 1995

A Charge Control and Current-Voltage Model for Inverted MODFET's A. F. M. Anwar, K.-W. Liu, and A. N. Khondker, Member, ZEEE

Absbuct- An analytical model is used to investigate proper- AlGaAs 4 1 * GaA5 F2, " 4 2

/ ties of the two-dimensional electron gas (2DEG) confined in a GaAdAIGaAs quantum well (Qw) formed in a inverted modula- tion doped field effect transistor (MODFET). The position of the Fermi level and the average distance of the carriers in the well have been calculated as a function of the 2DEG concentration, n,. A charge control model is presented based on the self-consistent solution of Schriidinger and Poisson's equation. The results show a unique behavior of the average distance of the 2DEG which increases with n,, a property unique to these type of structures. The analysis is extended to model current-voltage characteristics.

$ v , q b y e

4 . n l . 0 1

X

Undoped Doped x = O

(a)

I. INTRODUCTION ELENTLESS efforts to investigate devices that impart R enhanced performance have led to the development of

inverted MODFET (IMODFET) structures [ 11-[7], where the gate is placed on GaAs instead of on AlGaAs layer. The advantages of these structure are the following: 1) a more stable surface that facilitates a better ohmic contact to be established [l]; and 2) a higher transconductance than that of the normal structure [2]. The possibility of higher transconduc- tance has resulted in a renewed interest in inverted structures as is evident from the recent publications [6] , [7]. Though a number of works on charge control of these devices have been reported, all of them use a classical approach to the problem and to the best of our knowledge, no work on the quantum well (QW) characteristics have yet been performed. It was shown by Yoshida [8] that the charge control mech- anism of a normal MODFET can be adequately modeled by calculating the 2DEG concentration using 3D density of states and Fermi-Dirac probability distribution. This result may lead researchers in this field to believe that one can always use this procedure for IMODFET's as calculated by Lee et al. [l]. We have carried out self-consistent solution of Schrodinger and Poisson's equations to explore whether such a classical calculation can adequately describe the charge control mechanism. Details of the self-consistent solution method have been discussed elsewhere [9]. Furthermore, the QW properties, e.g., the variation of the average distance of the carriers from interface, z,, and the subband occupancy factors with the 2DEG concentration n,, have been investigated. The model

Manuscript received July 8, 1994; revised September 12, 1994. The review of this paper was arranged by Associate Editor N. Moll.

A. F. M. Anwar and K.-W. Liu are with the Department of Electrical and Systems Engineering, The University of Connecticut, Storrs, CT 06269-3 157 USA.

A. N. Khondker is with the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY 13676 USA.

IEEE Log Number 9409042.

001 8-9383/95$O4

Undoped Undoped Doped x = O

(b)

Fig. 1. Conduction band diagrams for two kinds of inverted MODFET's (IMODFET's) [l]. IMODFET I in (a) has a thick doped AlGaAs layer and IMODFET II in (b) has a thin doped AlGaAs layer.

presented in this paper is applicable to any inverted structures and for the sake of comparison with published results, a GaAs/AlGaAs system is considered.

11. CHARGE CONTROL A quantum well (QW) is formed at the interface between

AlGaAdGaAs and the electrons of the 2DEG are transferred from the donors in the AlGaAs layer to a quantum well (or channel) at the heterojunction. The eigen energies and the number of electrons in each subband depend on the shape of the quantum well as well as on the temperature. The conduction band diagram of the two types of device structures (IMODFET I and 11) are shown in Fig. l(a) and (b), respec- tively [ 13. The method used to solve Schrodinger and Poisson's equation self-consistently is similar to that used in solving for the QW formed at the normal MODFET [9]. The shape of the QW well formed in an inverted MODFET, as shown in Fig. 2, is approximated by straight lines with slopes al, a2, and a3 (dashed lines). In calculating the number of ionized dopants, a shallow impurity level was assumed, because we consider low doping density for which the contribution of deep donor levels may be ignored. In this study, both inverted MODFET structures (IMODFET I and 11) [l], [2] are considered.

.00 0 1995 IEEE

7- I'll I I

I I ,

ANWAR et al.: A CHARGE CONTROL AND CURRENT-VOLTAGE MODEL FOR INVERTED MODFET'S

ZDEC Concentration, n,= 1x10'~cm?

0.31

0.02

fi \

0.01 2 c

0.00

- 100 0 I00 200 300

Distance (A)

Fig. 2. The conduction band profile is plotted as a function of distance at 77 K. The eigen energies (dashed-dotted lines) and the distribution of 2DEG concentration n,(z) (dotted line) are also plotted. A 2DEG concentration ns = 1 x 10l2 cmP2 is assumed.

The electric field at the interface originating from the depleted AlGaAs layer, for the first type of structure, is written as [119 V I

F(0) = F ( d , ) = 2

where 6 = -k . T/q(ln(l + g'y) + 4/Nd h(l + y / 4 ) ) and y = exp(q . V ( - o o ) / k . T), k is the Boltzmann's constant, T is the temperature. g' = g[exp(Ed/k . T)] where g is the degeneracy factor of the donor level and Ed is the donor activation energy, Nd is the AlGaAs side doping density, di is the width of the spacer layer, A E , is the conduction band discontinuity and E and q are the permittivity and elemental electronic charge respectively. E,( - CO) = EFO - qcV( - CO ) , where EFO is the position of the Fermi level with respect to the tip of the conduction band at the interface and V(-CO) is the position of the conduction band with respect to the Fermi level found by the requirement of space charge neutrality. For the second type of structure the electric field at the heterointerface may be written as [l], [2]

where dd is the width of the doped AlGaAs layer. This relation for calculating the electric field is valid as long as the shallow impurity level is above the Fermi level, otherwise (1) must be used.

The slopes aj of the straight lines, which approximate the shape of the well, are proportional to the average electric field determined by Poisson's equation. The slopes may be written

587

(3)

where the factor fj's are determined by the self-consistent solution of Schrodinger and Poisson's equations as outlined by Khondker et al. [9], and may be interpreted as the fraction of the average 2DEG density to the left of the jth boundary.

To facilitate calculation the self-consistently calculated z,, and EF are expressed in the following functional forms (see Fig. 4)

xa,(ns(x)) = a + b . ln(ns(z)) (A) (4) E F ( ~ ~ ( ~ ) ) = E F O +r*ln(n,(z))

=EFO + AEF(ns(x)) (ev) ( 5 )

where x represents the position in the channel with respect to the source at x = 0. The numerical values of a, b, EFO and y are defined in Results and Discussion section.

Using reduced potentials s and p at the source and drain, respectively, the 2DEG concentration at the source can be written as [ l l ]

where

.S= V,, - A E F ( x = O ) / q - VT lvTl

V,, - V ( X = L1) - A E F ( z = L l ) / q - VT lvTl

P =

L1 is the length of unsaturated region of the channel. deR = d - Ad(n, (z = 0)) [l] and d is the thickness of GaAs layer, Ad(n,(x = 0)) denotes the effective channel width at the Source and equals (€AlGaAs/€GaAs).zav(ns(Z = 0)) [91, [111, [13]. It should be mentioned that A d ( n , ( z ) ) is a function of n,(z) and is properly updated depending on n,(z) (the channel is narrower at the source (z = 0) and widens as one move towards the drain region (z = L,)) [9], [ll]. Using (5) , (6) and the definition of s, the gate-source voltage may be written as

where the threshold voltage VT = (bb - q/t (Nddz/2 + Ndddd; + N ~ d ~ / 2 ) . $b,dd,d;, N A and Nd are the Schottky barrier height, the thickness of AlGaAs donor layer, the thick- ness of spacer layer, the unintentional doping concentration of GaAs layer and doping concentration of AlGaAs layer, respectively .

111. CURRENT-VOLTAGE CHARACTERISTICS Assuming ~d = ( U , . E ) / d ( ~ , / p o ) ~ + E 2 for the velocity-

electric field ( U d - E ) dependence, where Ud is the drift velocity, po is the low field mobility, U , is the saturation velocity and

_1 'I 1 ' 1 1

588

70-

h

5 i 60-

BEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4, APRIL 1995

- Inverted MODFET Type I .......... Inverted MODFET Type I1

0 3 6 9 12 15

2DEG Concenlration, 11, ( 10'' C I T I - ~ )

Fig. 3. IMODFET I at 300 K (solid lines) and 77 K (dashed lines).

Subband occupancy factors versus 2DEG concentration nsr for

E is the electric field, the saturation drain current reduced potential scheme can be written as [ l l ]

in the

The magnitude of the drain-source voltage for a given drain current is given by

. [1+{(yk-1)Yy}7 (9)

for 0 < I , < Ic,sat [12]. When the device is in saturation, IC > the drain-source voltage can be written as [12], ~ 4 1

A E F ( x 4 = L g ) ) ( + ? r (&

v d s = v, - VT -

- Ic(l/GO - Rd - R S ) 2 . deR . Eo

Here I , is the channel current, a = 3 / 2 , p = 2/3,&0 = w,/po, Go = (t . 2 . w,/d,R), L, and 2 are the gate length and width, respectively. R , is the source resistance and Rd is the drain resistance. To fit the experimental I-V curves in the saturation region, a parasitic resistance R p is introduced. The modified drain-source current Ids = I, + v d s / & , .

0.125

h > 0 . 1 0 0 ~

W d

al 3 cl

0.075 .i $. :

0.050

0 3 6 9 12

ZDEG Concentration, n, (10" cm-z)

Fig. 4. The averqge distance of 2DEG from the gate, zav, and the position of Fermi level, EF, are plotted as a function of 2DEG concentration n, for both type I (solid lines) and II (dashed lines) IMODFET's.

IV. RESULTS AND DISCUSSION Schrodinger and Poisson's equations are solved self-

consistently to determine an important parameter of the QW, i.e., the average distance of the 2DEG from the heterointerface as a function of the 2DEG density, n,. The calculations are performed at 77 K to compare with the classical results reported in Lee et al. [I]. Parameters used in this calculation are; m; = 0.067m0,m; = 0.0833m0, the conduction band discontinuity A E , = 0.3 eV, E = 13.le0, where EO = 8.854 x kg, Ed = 1 1 meV and the Schottky barrier height, 4b = 0.8 eV. For simplicity of analysis, the permittivity of both the layers are assumed to be the same. At the end of the calculation, the constants used in (4) and (5) are estimated as follows, x,,(n, =

-0.17(-0.19) and y = O.Ol(O.011) for IMODFET I (IMODFET 11) structure.

In Fig. 2, the self-consistently calculated potential profile is shown along with the eigen energies (dashed-dotted line). 2DEG concentration n, = 1 x lo1' cm-' is assumed. On the same plot the 2DEG distribution n,(x) (dotted line) is plotted as a function of distance. The nature of n, ( x ) can be explained in terms of the fractional occupancy factor as shown in Fig. 3. At 77 K (dashed lines) the first sub-band accommodates most of the carriers and this mean a better carrier confinement. Whereas at 300K (solid lines), the higher subbands are also populated and the confinement degrades slightly.

In Fig. 4 the average distance of the 2DEG, x,, , from the heterointerface and the position of Fermi level, E F , are plotted as a function of 2DEG concentration, n, for both IMODFET I (solid lines) and I1 (dashed lines). x,, for the inverted structure increases slowly as n, is increased, contrary to the behavior of the average distance for the normal MODFET's. This may be attributed to the placement of the gate, which is on GaAs and the application of a gate voltage shifts the 2DEG cloud towards the gate, thereby, increasing xau.

f/m and mo = 9.1 x

0 ) = -589.8(-939.8),b = 23.35(35.88) ,E~(n, = 0 ) =

1 1 ' I 1

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ANWAR et al.: A CHARGE CONTROL AND CURRENT-VOLTAGE MODEL FOR INVERTED MODFET’S 589

‘*I Inverted MODFETs Type I I

-1.0 -0.5 0.0 o s 1 .o Gate Voltage ( V )

Fig. 5. 2DEG concentration ns is plotted as a function of gate voltage for IMODFET I with d, = 608,. The width of the GaAs layers are 300 8, (solid line), 500 8, (long dashed line) and 700 8, (short dashed line), respectively. The diamonds, squares, and triangles represent classical results [I].

12- Invertcd MODFETs Type I1

d , = F O A

i -r - 1 0 -0 5 0 0 0 5 1 0

Gat r Voltage ( V )

Fig. 6. 2DEG concentration ng is plotted as a function of gate voltage for IMODFET I1 with d , = 608,. The width of the GaAs layers are 300 8, (solid line), 500 8, (long dashed line) and 700 8, (short dashed line), respectively. The diamonds, squares, and triangles represent classical results [I].

In Fig. 5 the 2DEG density, n,, is plotted as a function of the gate voltage, VG, for IMODFET-I. The width of the GaAs layer is varied from 300 A to 700 A at an interval of 200 A. The spacer layer thickness, di = 60 A. The comparison of the quantum mechanical and classical results [l] show a marked overestimation in the value of n, at lower 2DEG concentration and the difference decreases as the value of n, is increased. Also the linearity of the curve in the low concentration region reflects the fact that the value of n, is very small and the voltage drop is mainly in the unintentionally doped GaAs layer. In the n, - VG calculation, we have assumed a zero

1.8

h

4 1.5 E v

‘II ; 1.2

0 2 0.9 B U

0.6

I d

2 0.3

0.0

D.5x10pmz Inverted MODFET Type II

36 = 806, d = 40.4

Experimental Data :

V = 0.6V

V = 0.5V 6 . --

Vn 0.3 V --- -

?-

I I I I I

0 0 05 I O 1 5 2 0

Drain-Sourcr Vnllagr, Vds(V)

Fig. 7. Current-voltage (I-V) characteristic is plotted for a 0.5 x 10pm2 IMODFET II with d d = 808, and d , = 408,. The dots represent experimental data [5].

gate leakage current. For this reason we have restricted our calculations to values of n, much less than Q N ~ , where Q N ~ = (E /q )F(d ; ) , which ensures that the electric field in the GaAs layer is always greater than zero. We recognize that in real IMODFET devices the assumption of zero gate current is not strictly valid and as a result, our model is only valid as long as the gate current is negligible. We mention, however, that when the gate current becomes appreciable, the IMODFET loses its attractiveness as a useful device. Fig. 6 is plotted with d d = 100 A and di = 60 8, for IMODFET 11. A similar behavior of the curves, as discussed in this paragraph, is observed.

In Fig. 7, current-voltage (I-V) characteristics are plotted for a 0.5 x 10 pm2 IMODFET I1 structure with d d = 80 8, and di = 40 A. The plots are obtained using a calculated threshold voltage of 0.1 V. The low-field mobility and saturation velocity used in the calculation are po = 5800 cm2V-’sec-’ and w, = 0 . 9 ~ lo7 cmsec-’. To fit the experimental I-V curves in the saturation region, a parasitic resistance Rp is intro- duced. The source and drain resistances are 10 R and 10 R , respectively, the parasitic resistance Rp is 17 KR. On the same plot experimental data [5] are shown and reflects an excellent agreement.

V. CONCLUSION A self-consistent solution is presented to determine the

quantum mechanical properties of two types of inverted GaAs/AlGaAs MODFET’s. The calculation of x,, and EF as a function of n, is used to model their charge control and current-voltage characteristics. It seems that the self- consistently calculated charge control along with the I-V calculation are needed to explain experimental results [5].

REFERENCES

[l] K. Lee and M. Shur, “Charge control model of inverted GaAdAIGaAs modulation doped FET’s (IMODFET’s),” J. Vac. Sci. Technol., vol. B2, no. 2, Apr./June 1984.

590 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4, APRIL 1995

121 N. C. Cirillo, M. S. Shur, and J. K. Abrokwah, “Inverted GaAs/AIGaAs modulation doped field-effect transistors with extremely high transcon- ductances,” IEEE Electron Device Lett., vol. EDL-7, no. 2, pp. 71-74, 1986.

[3] D. Kim, A. Madhukar, K.-Z. Hu, and W. Chen, “Realization of high mobilities at ultralow electron density in GaAs/Alo ,Gao 7As inverted heterojunctions,” Appl. Phys. Lett., vol. 56, no. 19, pp. 1874-1876, 1990.

[4] P. M. Echtemach, K. Hu, A. Madhukar, and H. M. Bozler, “Transport measurement on a high mobility, ultralow camer concentration inverted GaAdAlGaAs heterostructure,” Physica E , vols. 165 and 166, pp. 871-872, 1990.

[SI S. Nishi, S. Seki, T. Saito, H. 1. Fujishiro, and Y. Sano, “A 0 .5 -p -ga te GaAdAIGaAs inverted HEMT IV-multiplier and D/A converter,” IEEE Trans. Electron Devices, vol. 36, no. 10, pp. 2191-2195, 1989.

[6] A. E. Schmitz, L. D. Nguyen, A. S. Brown, and R. A. Metzger, “InP- based inverted high electron mobility transistors,” IEEE Trans. Electron Devices, vol. 38, no. 12, p. 2702, 1991.

171 M. Kasashima, Y. Arain, H. 1. Fujishiro, H. Nakamua, and S. Nishi, “Pseudomorphic inverted HEMT suitable to low supplied voltage ap- plication,” IEEE Trans. Microwave Theory and Technol., vol. MTT-40, no. 12, pp. 2381-2386, 1992.

[8] J. Yoshida, “Classical versus quantum mechanical calculation of the electron distribution at the n-AIGaAs/GaAs heterointerface,” IEEE Trans. Electron Devices, 33, pp. 15&156, 1986.

[9] A. N. Khondker and A. F. M. Anwar, “Analytical model for Al- GaAs/GaAs heterojunction quantum wells,” Solid-State Electron., vol. 30, no. 8, pp. 847-852, 1987.

[ IO] A. F. M. Anwar and A. N. Khondker, “An envelope function description of double-heterojunction quantum wells,” .I. Appl. Phys., vol. 62, no. IO, p. 4200, 1987.

[ 1 11 A. F. M. Anwar and Kuo-Wei Liu, “Noise properties in AlGaAsKaAs MODFETs,” IEEE Trans. Electron Devices, vol. 40, no. 6, pp. I 1 7 6 1 176, June 1993.

[ 121 A. N. Khondker and A. F. M. Anwar, “Approximate analytic current- voltage calculation for MODFET’s,” IEEE Trans. Electron Devices, vol. 37, p. 314-317, 1990.

[13] A. N. Khondker, A. F. M. Anwar, M. A. Islam, L. Limoncelli, and D. Wilson, “Charge control model for MODFETs,” IEEE Trans. Electron Devices, vol. 33, pp. 1825-1826, 1986.

[ 141 A. B. Grebene and S. K. Ghandi, “General theory for pinched operation of the junction-gate FETs,” Solid-State Electron., vol. 12, pp. 573-589, 1969.

A. F. M. Anwar was bom in Jessore, Bangladesh. He received the B.S. degree in 1982 and the M.S. degree in 1984, both in electrical and electronic engineering from the Bangladesh University of En- gineering and Technology, Dhaka, Bangladesh. He received the Ph.D. degree in electrical engineering from Clarkson University, Potsdam, NY, in 1988.

He is now an Associate Professor in the Depart- ment of Electrical and Systems Engineering at the University of Connecticut, Storrs, CT. His research interests include transport in lower dimensional de-

vices and study of noise in HEMT’s, HBT’s, QW lasers, and other quantum size effect devices. He is also active in the growth and fabrication of Type I1 GaInSbhAs-based quantum well far infrared detectors.

K.-W. Liu received the B S.EE degree from the Ocean University, Keelung, Taiwan, in 1986, the M.S.E.E. degree from Syracuse Univerqity, Syracuse, NY, in 1990, and the Ph.D. degree in electrical engineering from the University of Connecticut, Storrs, CT, in 1994 His P h D dissertation was on the studv of noise in HEMT’s

He is presently a research engineer in the Op- toelectronics and Systems Laboratories, Industrial Technology Research Institute (ITRI), Chutung, Hsinchu, Taiwan, ROC. His research interests

include device simulation, fabrication and noise performance of MESFET’s, HEMT’s, HBT’s, and lasers.

A. N. Khondker (M’83) was born in Bangladesh. He received the B.Sc. degree in 1978 and the MSc. degree in 1980, both in electrical engineering from the Bangladesh University of Engineering and Technology, Dhaka. He received the Ph.D. degree in electrical engineering from Rice University, Houston, TX, in 1983.

He is currently an Associate Professor in the Department of Electrical and Computer Engineering at Clarkson University, Potsdam, NY. His research interests include semiconductor thin-film devices, modeling of compound semiconductor devices, and transport theory of mesoscopic structures.

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