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Improved analytic MODFET charge-control model
0038-I101/93 $6.00+ 0.00 Copyright 0 1993Pergamon Press Ltd
Solid-State Electronics Vol. 36, No. 3, pp. 481-482, 1993 Printed in Great Britain. All rights reserved
NOTE IMPROVED ANALYTIC MODFET CHARGE-CONTROL (Received
27 May
1992: in revised form
8 September
MODEL
1992)
Weak inversion
INTRODUCTION Closed-form expressions are proposed for charge control in modulation-doped FETs. They model operation in the strong, weak and moderate inversion regimes. These expressions are derived as limiting forms of the governing transcendental equations and hence are entirely physically based. The expressions compare favourably with numerical computation of the governing equations and are ideally suited for application in circuit simulation programs.
At low interface carrier concentrations, EF lies below the first two energy levels (E, and E,) and the exponential terms are very small. We can now expand the logarithmic term in the density of states expression to first order to give:
DISCUSSION
At extremely small sheet-carrier concentrations (corresponding to weak inversion), both exponential terms tend to 1 and hence:
The modulation-doped FET (MODFET) is a candidate for high-speed circuit applications. In its simplest form, it consists of an abrupt junction between materials with differing electrons affinities and work-functions, with the larger bandgap material being doped and the smaller bandgap material being undoped. A two-dimensional electron gas forms in the smaller bandgap region which can then be subjected to an electric field in a direction orthogonal to the controlling (gate) voltage, giving rise to standard transistor action. The area1 charge density is calculated by simultaneously solving Poisson’s equation, the Schrodinger equation and the density of states equation[l]. Poisson’s equation gives us[l]:
-!L 2Dk, T
Substituting (3) in (la), we have: n,+?
Vrln __ [ 2Dk, T In weak inversion the logarithmic hence:
it,=?
where n3 is the area1 charge density in the channel, t, is the permittivity and d the thickness of the larger bandgap material (typically AlGaAs; region 1 in Fig. l), q is the electronic charge, Vo is the gate voltage and V,,, is a voltage determined by the Schottky barrier height, d and the doping concentration in the AlGaAs. EF is the Fermi level as defined in Fig. 1. Note that expression (la) assumes that all donors in region 1 are ionized. The density of states is:
term dominates and
Strong inversion At very large sheet carrier concentrations, where the Fermi level is several k, T above the bottom of the potential well, the exponential terms in (1b) become much larger than 1 and hence:
(lb)
where D = m*/nh2, E, = y,ns2” and the yi are chosen to fit experimental data (typically, only the first two energy levels are of importance). The ,I?,- n, relationship is derived by assuming an infinite, triangular potential well. In general, only a numerical solution to eqns (1a) and (1b) is possible. Approximate solutions have been proposed in the literature for the relationship between Er and n,, including a quadratic relationship between the two quantities[2], an approximation involving n,‘j3[3], and by adding a n,-’ term to a linear approximation[4]. All these modelling efforts have perforce involved the use of fitting parameters. In this paper, we show that it is possible to derive approximate, analytic, closed-form solutions to the above equations without resorting to fitting parameters. These solutions correspond to the device being in weak inversion, moderate inversion and strong inversion and are hence easily usable in modelling the output characteristics of the three-terminal MODFET for circuit simulation purposes[5,6].
(4)
Note that although this expression is valid only for a two-dimensional electron gas, a similar exponential dependence may be derived for a three-dimensional gas (as is the case for the MOSFET). The MODFET electron gas tends to be quasi-3D in this regime, and hence the above expression is only approximately true.
qd
n~=Dk.~ln[~,{l+exp(!$$)}].
[ 1. V,,,). qd “s1“$V,E,=k,Tln
Using (6) in (la) we have
(1 +&)+n?&
(YO+h)~;(VG-
vOFFh
(7) Above threshold, the linear term dominates the term in n:/r and hence (7) reduces to ?J,z
,
61
\ (Vo - Vorr).
(8)
@(I‘j&J Since we have assumed complete ionization of donors, this expression is only valid until the formation of the parasitic MESFET in region 1 due to free electron generation and simultaneous donor neutralization.
481
482
Note Since we have assumed Er 6 k, Tand the second term on the right-hand side is several k,T in the region of interest, we can express the Fermi level as:
where nV, = qdn,/c,.
d Fig.
1. Energy
band
diagram of structure.
a typical
MODF ‘ET
Moderare inversion In the transition region between the pure exponential dependence of (5) and the pure linear dependence of (8), an approximate analysis is used to obtain the n,-Po relationship. When the Fermi level is at the bottom of the potential well in region 2 (i.e. Er = 0), we can consider the device to be just below threshold. Equation (2) is still a valid firstorder approximation and may be rewritten as: n, = Dk, T{exp(g)
+ exp(*)}exp(&). (9)
Equation (9) can be solved iteratively to obtain is defined as n, at .I+ = 0. We assume that increments of Er: ns=ndexp($)-rr,(l+s). Using
(10) in (la),
n,, which for small
(10)
we have:
Using
ns 1 na exp
(12) in (10) we have:
vo- vow-nv,
(
“Vr
The threshold voltage may now be defined as VTH= V,,,, + n V,. Figure 2 compares the approximations of eqns (5) (8) and (13) with the iterative solution to eqns (la) and (1 b). Parameters used in the calculation are d=400& V0FF = 0 V, T = 300 K, M * = 0.07 m, (where nr. is the electron mass) as in GaAs, and c, = 1.062 IO-” Fm-‘. The calculated threshold voltage, as defined above, is 0.145 V. As is evident from the results, the approximations derived in this paper compare very favourably with the numerical results. The equations are ideally suited for application in circuit simulation programs since they explicitly express the charge density in terms of the gate voltage. They are based entirely on physical parameters and do not require any curve-fitting. SUMMARY
A three-piece charge-control model has been derived for MODFETs which covers the three principal regions of operation-weak, moderate and strong inversion. The analytic model compares well with numerical computation of eqns (la) and (lb). The major advantage of the proposed model is that it can be easily implemented in circuit analysis programs. Acknowledgemenf-This work was supported under contract 001-86-C-0062.
a
.El 5 2 -K
Field of Electrical Engineering Cornell University Ithaca, NY 14850, U.S.A.
10’6 lo15 1014 1
xx
:-.;i”, , , _
-0.40
0.05
-0.18
Gate
voltage
0.27
0.50
(V)
Fig. 2. Interface carrier concentration as a function of gate voltage. Continuous line-iterative solution to (la) and (1 b); squares-approximation of (5); crosses-approximation of (13); circles-approximation of (8).
(13)
>-
Departmenr of Electrical and Computer Engineering North Carolina State University Raleigh, NC 27695, U.S.A.
by the DNA
G. GEORGE
J. R. HAUSER
REFERENCES
and N. T. Linh, IEEE Trans. ED1. D. Delagebeaudeuf 29, 955 (1982). 2. S. Kola, J. M. Goho and G. N. Maracas, IEEE Trans. 9, 136 (1988). 3. M. J. Moloney, F. Ponse and H. Morkoc, IEEE Trans. ED-32, 1675 (1985). 4. K. Y. Tong, Electron. Left. 27, 668 (1991). 5. G. George and J. R. Hauser, IEEE Trans. ED-37, 1193 (1990). State Univ 6. G. George, M. S. thesis, North Carolina (1989).
Solid-State Electronics Vol. 36, No. 3, pp. 481-482, 1993 Printed in Great Britain. All rights reserved
NOTE IMPROVED ANALYTIC MODFET CHARGE-CONTROL (Received
27 May
1992: in revised form
8 September
MODEL
1992)
Weak inversion
INTRODUCTION Closed-form expressions are proposed for charge control in modulation-doped FETs. They model operation in the strong, weak and moderate inversion regimes. These expressions are derived as limiting forms of the governing transcendental equations and hence are entirely physically based. The expressions compare favourably with numerical computation of the governing equations and are ideally suited for application in circuit simulation programs.
At low interface carrier concentrations, EF lies below the first two energy levels (E, and E,) and the exponential terms are very small. We can now expand the logarithmic term in the density of states expression to first order to give:
DISCUSSION
At extremely small sheet-carrier concentrations (corresponding to weak inversion), both exponential terms tend to 1 and hence:
The modulation-doped FET (MODFET) is a candidate for high-speed circuit applications. In its simplest form, it consists of an abrupt junction between materials with differing electrons affinities and work-functions, with the larger bandgap material being doped and the smaller bandgap material being undoped. A two-dimensional electron gas forms in the smaller bandgap region which can then be subjected to an electric field in a direction orthogonal to the controlling (gate) voltage, giving rise to standard transistor action. The area1 charge density is calculated by simultaneously solving Poisson’s equation, the Schrodinger equation and the density of states equation[l]. Poisson’s equation gives us[l]:
-!L 2Dk, T
Substituting (3) in (la), we have: n,+?
Vrln __ [ 2Dk, T In weak inversion the logarithmic hence:
it,=?
where n3 is the area1 charge density in the channel, t, is the permittivity and d the thickness of the larger bandgap material (typically AlGaAs; region 1 in Fig. l), q is the electronic charge, Vo is the gate voltage and V,,, is a voltage determined by the Schottky barrier height, d and the doping concentration in the AlGaAs. EF is the Fermi level as defined in Fig. 1. Note that expression (la) assumes that all donors in region 1 are ionized. The density of states is:
term dominates and
Strong inversion At very large sheet carrier concentrations, where the Fermi level is several k, T above the bottom of the potential well, the exponential terms in (1b) become much larger than 1 and hence:
(lb)
where D = m*/nh2, E, = y,ns2” and the yi are chosen to fit experimental data (typically, only the first two energy levels are of importance). The ,I?,- n, relationship is derived by assuming an infinite, triangular potential well. In general, only a numerical solution to eqns (1a) and (1b) is possible. Approximate solutions have been proposed in the literature for the relationship between Er and n,, including a quadratic relationship between the two quantities[2], an approximation involving n,‘j3[3], and by adding a n,-’ term to a linear approximation[4]. All these modelling efforts have perforce involved the use of fitting parameters. In this paper, we show that it is possible to derive approximate, analytic, closed-form solutions to the above equations without resorting to fitting parameters. These solutions correspond to the device being in weak inversion, moderate inversion and strong inversion and are hence easily usable in modelling the output characteristics of the three-terminal MODFET for circuit simulation purposes[5,6].
(4)
Note that although this expression is valid only for a two-dimensional electron gas, a similar exponential dependence may be derived for a three-dimensional gas (as is the case for the MOSFET). The MODFET electron gas tends to be quasi-3D in this regime, and hence the above expression is only approximately true.
qd
n~=Dk.~ln[~,{l+exp(!$$)}].
[ 1. V,,,). qd “s1“$V,E,=k,Tln
Using (6) in (la) we have
(1 +&)+n?&
(YO+h)~;(VG-
vOFFh
(7) Above threshold, the linear term dominates the term in n:/r and hence (7) reduces to ?J,z
,
61
\ (Vo - Vorr).
(8)
@(I‘j&J Since we have assumed complete ionization of donors, this expression is only valid until the formation of the parasitic MESFET in region 1 due to free electron generation and simultaneous donor neutralization.
481
482
Note Since we have assumed Er 6 k, Tand the second term on the right-hand side is several k,T in the region of interest, we can express the Fermi level as:
where nV, = qdn,/c,.
d Fig.
1. Energy
band
diagram of structure.
a typical
MODF ‘ET
Moderare inversion In the transition region between the pure exponential dependence of (5) and the pure linear dependence of (8), an approximate analysis is used to obtain the n,-Po relationship. When the Fermi level is at the bottom of the potential well in region 2 (i.e. Er = 0), we can consider the device to be just below threshold. Equation (2) is still a valid firstorder approximation and may be rewritten as: n, = Dk, T{exp(g)
+ exp(*)}exp(&). (9)
Equation (9) can be solved iteratively to obtain is defined as n, at .I+ = 0. We assume that increments of Er: ns=ndexp($)-rr,(l+s). Using
(10) in (la),
n,, which for small
(10)
we have:
Using
ns 1 na exp
(12) in (10) we have:
vo- vow-nv,
(
“Vr
The threshold voltage may now be defined as VTH= V,,,, + n V,. Figure 2 compares the approximations of eqns (5) (8) and (13) with the iterative solution to eqns (la) and (1 b). Parameters used in the calculation are d=400& V0FF = 0 V, T = 300 K, M * = 0.07 m, (where nr. is the electron mass) as in GaAs, and c, = 1.062 IO-” Fm-‘. The calculated threshold voltage, as defined above, is 0.145 V. As is evident from the results, the approximations derived in this paper compare very favourably with the numerical results. The equations are ideally suited for application in circuit simulation programs since they explicitly express the charge density in terms of the gate voltage. They are based entirely on physical parameters and do not require any curve-fitting. SUMMARY
A three-piece charge-control model has been derived for MODFETs which covers the three principal regions of operation-weak, moderate and strong inversion. The analytic model compares well with numerical computation of eqns (la) and (lb). The major advantage of the proposed model is that it can be easily implemented in circuit analysis programs. Acknowledgemenf-This work was supported under contract 001-86-C-0062.
a
.El 5 2 -K
Field of Electrical Engineering Cornell University Ithaca, NY 14850, U.S.A.
10’6 lo15 1014 1
xx
:-.;i”, , , _
-0.40
0.05
-0.18
Gate
voltage
0.27
0.50
(V)
Fig. 2. Interface carrier concentration as a function of gate voltage. Continuous line-iterative solution to (la) and (1 b); squares-approximation of (5); crosses-approximation of (13); circles-approximation of (8).
(13)
>-
Departmenr of Electrical and Computer Engineering North Carolina State University Raleigh, NC 27695, U.S.A.
by the DNA
G. GEORGE
J. R. HAUSER
REFERENCES
and N. T. Linh, IEEE Trans. ED1. D. Delagebeaudeuf 29, 955 (1982). 2. S. Kola, J. M. Goho and G. N. Maracas, IEEE Trans. 9, 136 (1988). 3. M. J. Moloney, F. Ponse and H. Morkoc, IEEE Trans. ED-32, 1675 (1985). 4. K. Y. Tong, Electron. Left. 27, 668 (1991). 5. G. George and J. R. Hauser, IEEE Trans. ED-37, 1193 (1990). State Univ 6. G. George, M. S. thesis, North Carolina (1989).