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Estimation of the accuracy of the microwave FET equivalent circuit using the quasi-hydrodynamic model of electron transport
Solid-State Electronics 103 (2015) 115–121
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Solid-State Electronics journal homepage: www.elsevier.com/locate/sse
Estimation of the accuracy of the microwave FET equivalent circuit using the quasi-hydrodynamic model of electron transport Gennadiy Z. Garber Pulsar R&D Manufacturing Company, Moscow 105187, Russia
a r t i c l e
i n f o
Article history: Received 7 April 2014 Received in revised form 6 August 2014 Accepted 15 August 2014 Available online 11 September 2014 The review of this paper was arranged by Prof. S. Cristoloveanu Keywords: Equivalent circuit FET Frequency-domain simulation Quasi-hydrodynamic model Spline characteristics Time-domain simulation
a b s t r a c t Computational methods are considered for the time- and frequency-domain simulations of a microwave power amplifier with a transistor built on a diamond-like semiconductor. The first method is based only on a two-dimensional model of the intrinsic transistor that includes the quasi-hydrodynamic model of electron transport and electric field equations. The second method, which is less accurate but much more computer-time-saving, is based on the intrinsic transistor large-signal lumped-element equivalent circuit model with the parameters and spline characteristics calculated by means of the abovementioned two-dimensional model. Using these two methods, the microwave parameters of the amplifier are calculated for frequencies varying from 15 to 90 GHz. When using the equivalent circuit at frequencies of up to 60 GHz, the calculation error (equal to the difference between the results of the time- and frequency-domain simulations) does not exceed 5% for the input impedance and 0.6 dB for the transducer power gain. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Currently, field effect transistors (FETs) are widely used as the active elements of power amplifiers in the frequency range from several hundred megahertz to several hundred gigahertz [1–3]. Therefore, the adequacy of the intrinsic FET (IFET) large-signal lumped-element equivalent circuit model (Fig. 1) used for designing these amplifiers is becoming a key issue. In [4–6], we developed a method for calculating the characteristics and parameters of the IFET equivalent circuit that is based on the cubic spline construction [7] and the twodimensional (2D) quasi-hydrodynamic model of electron transport [8–11]. Below, we will calculate a number of microwave parameters of a power amplifier based on a transistor that ‘‘is made’’ on an n-type semiconductor, in which the field dependence of the drift velocity of electrons is close to the field dependence of the drift velocity of holes in boron-doped diamond [1,12–14]. The physical parameters of the semiconductor and insulator (Fig. 2) are close to the corresponding constants for diamond.
The microwave parameters will be calculated in two ways: using only the 2D model of the IFET that includes the quasihydrodynamic model of electron transport and electric field equations and combining the 2D model with the IFET equivalent circuit. 2. 2D model of the intrinsic transistor Under the operating conditions of the transistor, we can neglect the impact ionization of semiconductor molecules by electrons and holes. Then, the 2D quasi-hydrodynamic model of electron transport takes the following form:
@n ¼ r~jn ; @t ~jn ¼ l ! n E n þ rðDn nÞ;
http://dx.doi.org/10.1016/j.sse.2014.08.008 0038-1101/Ó 2014 Elsevier Ltd. All rights reserved.
ð2Þ
! @ðnuÞ u u0 ¼ r~jnu þ q~jn E n ; @t sn ~jnu ¼ n l ! E nu þ rðn Dn nuÞ; n
E-mail address: [email protected] URL: http://www.gzgarber.narod.ru
ð1Þ
n
n
ð3Þ ð4Þ
where u(x, y, t) = 3kTe/2 is the mean kinetic energy of electrons; x and y are the horizontal and vertical coordinates (Fig. 2); t is the time; k is the Boltzmann constant; Te(x, y, t) is the electron gas
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121
Fig. 2. The active region of the delta-doped MISFET on the diamond-like semiconductor. Fig. 1. Large-signal equivalent circuit of the IFET without impact ionization of semiconductor molecules by electrons and holes: ich is the channel current generator; Cgs and Rgs, the input capacitance and resistance; and Cgd, the transfer (feedback) capacitance.
temperature; r ¼
@ @ ; @x @y
The field dependence of the electron drift velocity appearing in (7) is given by
V n ¼ v e ½1 expðle Eus =v e Þ; 3
is the gradient operator; u0 = 3kT0/2 is the
mean kinetic energy of electrons in thermodynamic equilibrium; T0 is the lattice temperature (in our case, T0 is set equal to 300 K); n(x, y, t), ln(x, y, u), Dn(x, y, u) = (kTe/q)ln, sn(x, y, u), and nn(x, y, u) are the concentration, mobility, diffusion coefficient, energy relaxation time, and relative Peltier coefficient of electrons, jn ðx; y; tÞ and respectively; q is the positive elementary charge; ~ jnu ðx; y; tÞ are the electron flux density and electron energy flux ~ ! ! density, respectively; E ðx; y; tÞ is the electric field intensity; ln E ! is the electron drift velocity; and E ¼ j E j. ! Electric field intensity E ðx; y; tÞ is calculated using the electric field equations
!
rðe E Þ ¼ Q ;
ð5Þ
! E ¼ ru;
ð6Þ
where u(x, y, t) is the electric potential and e(x, y) is the absolute permittivity of the semiconductor or insulator; in our case, the relative permittivity is equal to 5.7 for both the semiconductor and insulator. Charge density Q is equal to zero in the insulator and to q(Nd n) in the semiconductor, where Nd(x, y) is the concentration of ionized donors. The kinetic coefficients for electrons are calculated by the formulas
ln ðx; y; uÞ ¼ V n ½x; y; Fðx; y; uÞ=Fðx; y; uÞ;
ð7Þ
sn ðx; y; uÞ ¼ T n ½x; y; Fðx; y; uÞ;
ð8Þ
nn ðx; y; uÞ ¼ K n ½x; y; Fðx; y; uÞ;
ð9Þ
where Vn(x, y, Eus), Tn(x, y, Eus), and Kn(x, y, Eus) are the given field dependences of the drift velocity, energy relaxation time, and relative Peltier coefficient, respectively, and Eus is the intensity of the uniform stationary electric field. When finding the field dependences, we assume that the semiconductor is homogeneous and its chemical composition is the same as at a point with coordinates x and y (in the semiconducting part of the transistor active region depicted in Fig. 2). Quantity Vn(x, y, Eus) is the drift velocity of electrons in such a semiconductor subjected to uniform stationary electric field Eus. Quantities Tn(x, y, Eus) and Kn(x, y, Eus) have the same meaning. Function F(x, y, u) involved in (7)–(9) is zero at u 6 u0 and is a solution to the energy balance equation,
qFV n ðx; y; FÞT n ðx; y; FÞ ¼ u u0 ;
ð10Þ
at u > u0. This equation follows from (2) and (3) at r = o/ot = 0.
2
ð11Þ 15
3
2
where le = 2 10 cm /(V s) at Nd = 10 cm and 500 cm /(V s) at Nd = 1018 cm3 and ve = 1.1 107 cm/s. The field dependence of the energy relaxation time in (8) has the following form: function Tn(x, y, Eus) is equal to vese/Vn(x, y, Eus) for Eus > Ee and to vese/ Vn(x, y, Ee) for Eus 6 Ee, where se = 1 ps and Ee = 3ve/le. With function Tn(x, y, Eus) taken in such a form, the abovementioned solution to (10) at u > u0 is unique. The field dependence of the relative Peltier coefficient in (9) is given by
Kn ¼
2 expðEus =Ee Þ þ 1: 3
System (1)–(6) is considered in the transistor active region depicted in Fig. 2. Boundary conditions are the following: n = Nd and Te = T0 at the source and drain contacts; ! jn , ~ jnu , and E is equal to zero at the the normal component of ~ contact-free semiconductor parts of the boundary; ! the normal component of E equals zero at the contact-free insulator parts; the electric potentials of the source, gate, and drain contacts (us, ug, and ud, respectively) are given constants (when we calculate the characteristics and parameters of the IFET equivalent circuit) or are determined by the extrinsic electrical circuit (when we simulate the transistor amplifier without using the IFET equivalent circuit). At the insulator–semiconductor interface, we require that only the normal component of ~ jn and ~ jnu be zero (because the permittivity of the semiconductor and insulator is one and the same). In addition to the formulated boundary conditions, initial conditions are necessary. They are arbitrary or are determined by the 2D distributions of u, n, and Te, which result from the previous solution of (1)–(6). For solving the above initial-boundary value problem, we solve at each time step Eqs. (5) and (6) by the method of matrix elimination [15] and then Eqs. (1) and (2) and, concurrently, (3) and (4) using the locally one-dimensional scheme [15] with jnx, jny, jnux, and jnuy expressed as the electron current density in [16]. Then we calculate the source, gate, and drain currents:
is ¼
ZZ
source
ig ¼
ZZ
q jny þ e
@ Ey dxdz; @t
@ e Ey dxdz; @t gate
ð12Þ ð13Þ
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121
Fig. 3. Grid on the gate bias – drain bias plane.
id ¼
ZZ
@ q jny e Ey dxdz; @t drain
ð14Þ
where z is the third axis directed perpendicularly to the plane of Fig. 2. In our case, the gate width along the z axis is equal to 1 mm.
Fig. 4. Output current–voltage characteristics (continuous lines) and 90-GHz cycles calculated without using (dashed cycle) and using (dash-and-dot cycle) the IFET equivalent circuit.
3. Equivalent circuit of the intrinsic transistor Calculation of the characteristics and parameters of the equivalent circuit depicted in Fig. 1 starts with constructing a rectangular grid on the plane with axes Ugs = ug + ub us and Uds = ud us (Fig. 3), where ub = 5 V is the built-in potential of the structure depicted in Fig. 2. This grid is uniform in Ugs and nonuniform in Uds; Ugs,i and Uds,j are the nodal coordinates; i = 0, 1, ..., m; j = 0, 1, ..., n; Uds,0 = 0. The values of Ugs,0, Ugs,m, and Uds,n must be such that the grid entirely covers the operating region of the IFET on the Ugs–Uds plane. At each node of the grid, we solve the initial-boundary value problem for (1)–(6). The 2D distributions of the electric potential, electron concentration, and electron gas temperature at t = 1, u(x, y) = u(x, y, 1), n(x, y) = n(x, y, 1), and Te(x, y) = Te(x, y, 1), respectively, are filed. This file also contains the time dependences of the following physical characteristics: the source, gate, and drain currents – is(t), ig(t), and id(t); the electrostatic induction vector flux to the gate – qg(t) = RR gate e Ey dxdz (it has the dimension of charge). The dynamic transconductance and dynamic conductance describing the channel current generator (ich in Fig. 1) are calculated as
SðxÞ ¼
GðxÞ ¼
sinðxsÞ
xs sinðxsÞ
xs
expðjxsÞS0 ;
ð15Þ
expðjxsÞG0 ;
ð16Þ
where S0 = oId/oUgs and G0 = oId/oUds are the static transconductance and static conductance; Id(Ugs, Uds) is the dependence of id(1) = Id on Ugs and Uds; x is the circular frequency; j is the imaginary unit; s = sch/2; and sch is the time of electron transit through the channel’s nonohmic part. Note that Id = Is = is(1) since the gate current at t = 1 is absent. Formulas (15) and (16) taking into account the electron transit through the channel’s nonohmic part correspond to the following time-domain expression for the channel current:
ich ðtÞ ¼
1
sch
Z
t
0
Id ½ugs ðt 0 Þ; uds ðt0 Þdt ;
tsch
where ugs(t) = ug(t) + ub us(t) and uds(t) = ud(t) us(t).
Fig. 5. Charge–voltage characteristics (continuous lines) and 90-GHz cycles calculated without using (dashed cycle) and using (dash-and-dot cycle) the IFET equivalent circuit.
The dependence of qg(1) = Qg on Ugs and Uds written as Qg(Ugs, Uds) determines the input and transfer capacitances according to the formulas Cgs(Ugs, Uds) = oQg/oUgs oQg/oUds and Cgd(Ugs, Uds) = oQg/oUds, respectively. The resistance in the input branch is given by Rgs = sgs/Cgs, where sgs is the time constant of the resistor–capacitor network. The current in this branch is calculated by the formula based on the assumption that sgs is small in comparison with the oscillation period:
iin ðtÞ ¼ w0 ðtÞ þ sgs
dcgs w0 dw0 : dt cgs dt
ð17Þ
Here, cgs(t) = Cgs[ugs(t), uds(t)] and w0(t) = cgs(t) dugs/dt. The derivation of formula (17) is given in Appendix A. The current in the transfer branch is given by
itr ðtÞ ¼ cgd ðtÞ
dugd ; dt
where cgd(t) = Cgd[ugs(t), uds(t)] and ugd(t) = ug(t) + ub ud(t). We use the cubic spline construction [7] to analytically continue the discrete dependences found by solving (1)–(6) at the grid nodes on the Ugs Uds plane depicted in Fig. 3. Therefore, the dependences Id(Ugs, Uds) and Qg(Ugs, Uds) may be called the spline characteristics of the IFET equivalent circuit.
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121
are required parameters sgs and sch for the node (4.5; 10 V): sgs = 0.788 ps and sch = 1.229 ps. They are determined using the TIFET macro for Microsoft Office Excel. 4. Simulation of the amplifier in the time and frequency domains In the model amplifier depicted in Fig. 7, the operating point of the IFET has the coordinates Ugs = 4.63 V and Uds = 10.06 V, at which the transfer capacitance is equal to Cgd = 0.014 pF. For comparison, the corresponding value of the input capacitance is Cgs = 0.347 pF, i.e., the assumption that Cgd = 0 used to determine the time constants, sgs and sch, is valid. After using the TIFET macro for calculating sgs and sch in the nodes (5; 10 V), (5; 12 V), and (4.5; 12 V), which (as well as the node (4.5; 10 V) considered above) are the closest to the operating point, we obtain sgs = 0.794 ps and sch = 1.283 ps for the IFET operating point (4.63; 10.06 V) with linear interpolation. These values of sgs and sch are the parameters of the IFET equivalent circuit. We will simulate the amplifier (Fig. 7) in the time and frequency domains for a given amplitude and frequency of the electromotive force of the input RF generator. In what follows,
Fig. 6. Time dependences of the (a) gate and (b) drain currents (continuous lines) and their exponential approximations (dashed lines).
A method for calculating parameters sgs and sch of the equivalent circuit will be considered below. We developed a version of the DIHEMT program [10] to solve the initial-boundary value problem for (1)–(6) at the grid nodes. The continuous lines in Figs. 4 and 5 show the calculated characteristics Id(Ugs, Uds) and Qg(Ugs, Uds), respectively. One execution of the DIHEMT code makes it possible to simulate all nodes of one vertical line on the Ugs Uds plane (Fig. 3): i = 0, 1, ..., m; j = const. Under the assumption that Cgd = 0, parameters sgs and sch can be found from the time dependences of gate current ig(t) and drain current id(t) as follows. In Figs. 6a and b, continuous lines are the ig(t) and id(t) dependences calculated in going from the node (5; 10 V) to the adjacent node (4.5; 10 V) and dashed lines (calculated by the least-squares method) represent the exponential approximations of ig(t) and id(t). The time constants of these exponential curves are equal to 0.788 and 1.229 ps, respectively. According to [6], these values
Ld(ud, ug, us), Lg(ud, ug, us), and Ls(ud, ug, us) are, respectively, currents at points d, g, and s determined by a linear active three port (LATP), part of the circuit in Fig. 7 that is outside the IFET unit; Md(ud us, ug us), Mg(ud us, ug us), and Ms(ud us, ug us) = Md + Mg are currents at points d, g, and s determined by the IFET unit. When simulating the amplifier in the time domain, the solution of the initial-boundary value problem for (1)–(6) lasts until periodic oscillations (with period T) set in. At each time step, the quadratic discrepancy
Fðud ; ug ; us Þ ¼ ½Md ðud us ; ug us Þ Ld ðud ; ug ; us Þ2 þ ½Mg ðud us ; ug us Þ Lg ðud ; ug ; us Þ2 þ ½Ms ðud us ; ug us Þ Ls ðud ; ug ; us Þ2
ð18Þ
of the IFET drain, gate, and source currents is minimized, with Ms (ud us, ug us), Mg(ud us, ug us), and Md(ud us, ug us) calculated by (12)–(14). The electric potentials at points d, g, and s (relative to ground, Fig. 7) are adjustable parameters. We use the Powell method [17] to minimize (nearly to zero) function (18) at each time step. Note that not only functions Ld
Fig. 7. Electrical circuit of the model amplifier: resistors of 0.3 and 2 X are the parasitic elements of the transistor.
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121 Table 1 Calculation of the transistor input impedance. Frequency 1/T (GHz)
Power available from the RF generator (mW)
Simulation approach
Input voltage standing wave ratio
Real part of the input impedance (O)
Imaginary part of the input impedance (O)
Accuracy of input impedance calculation (%)
15
4.9
QHM only QHM & EC QHM & EC0
19.2 17.45 29.34
3.436 3.864 2.301
28.2 29.44 29.57
4.6 6.3
QHM only QHM & EC QHM & EC0
15.12 14.29 24.04
3.578 3.794 2.261
14.28 14.49 14.75
2 9.5
QHM only QHM & EC QHM & EC0
13.48 13.76 22.74
3.784 3.702 2.247
7.078 6.872 7.384
2.8 19.5
QHM only QHM & EC QHM & EC0
12.65 14.07 22.47
3.988 3.579 2.247
4.684 4.225 4.932
10 28.6
30
60
90
15.6
62.5
160
Table 2 Calculation of the transducer power gain of the amplifier. Frequency 1/T (GHz)
Power available from the RF generator (mW)
L (nH)
C (pF)
Simulation approach
Input voltage standing wave ratio
Output RF power (mW)
Transducer power gain (dB)
Accuracy of transducer power gain calculation (dB)
15
4.9
0.484
0.693
QHM only QHM & EC QHM & EC0
1.202 1.313 2.339
658.3 673.5 757.4
21.28 21.38 21.89
0.1 0.61
QHM only QHM & EC QHM & EC0
1.065 1.066 1.578
555 569.4 729.4
15.5 15.62 16.69
0.12 1.19
QHM only QHM & EC QHM & EC0
1.079 1.051 1.636
498.2 568.7 718.6
9.015 9.59 10.61
0.57 1.6
QHM only QHM & EC QHM & EC0
1.158 1.145 1.736
464.4 631.3 729
4.628 5.961 6.586
1.33 1.96
30
60
90
15.6
62.5
160
0.156
0.0572
0.0341
0.379
0.191
0.125
(ud, ug, us), Lg(ud, ug, us), and Ls(ud, ug, us) but also functions Md (ud us, ug us), Mg(ud us, ug us), and Ms(ud us, ug us) are linear. This assertion follows from the fact that, during minimization, we solve (5) with charge density Q taken from the previous time step. Since Ld(ud, ug, us), Lg(ud, ug, us), Ls(ud, ug, us), Md (ud us, ug us), Mg(ud us, ug us), and Ms(ud us, ug us) are linear, function (18) is quadratic, which facilitates the search for its minimum point. We developed a version of the DIHEMT program [10] for the time-domain simulation of the amplifier. The frequency-domain simulation, which is much more efficient in terms of CPU time saving, is based on the IFET equivalent circuit depicted in Fig. 1. We use the harmonic balance method (in our case, for five harmonics of the input RF generator frequency, 1/T) to obtain the periodic oscillation mode. To this end, the system of algebraic equations
M d ðud us ; ug us Þ ¼ Ld ðud ; ug ; us Þ;
ð19Þ
M g ðud us ; ug us Þ ¼ Lg ðud ; ug ; us Þ;
ð20Þ
M s ðud us ; ug us Þ ¼ Ls ðud ; ug ; us Þ
ð21Þ
is solved, where Md, Mg, Ms, Ld, Lg, and Ls are the complex amplitudes of currents at points d, g, and s determined by the IFET and LATP and ud, ug, and us are the complex amplitudes of potentials at points d, g, and s. Here, functions Md, Mg, and Ms are nonlinear. The system of nonlinear algebraic Eqs. (19)–(21) is solved by the Newton method with the Jacobian matrix including the derivatives of the cubic splines that describe the characteristics
of the IFET equivalent circuit. When analyzing this equivalent circuit, the direct and inverse discrete Fourier transformations [18] are performed. The Visual C++ program HIFETA (Harmonics In FET Amplifier) implements the frequency-domain simulation of the model amplifier. The calculated periodic time dependences of the currents and potentials (in both the time and frequency domains) govern the microwave parameters of the amplifier, some of which are given in Tables 1 and 2. Below, QHM stands for 2D model (1)–(6) and EC stands for the IFET equivalent circuit. The simulation frequencies are 1/T = 15, 30, 60, and 90 GHz. 5. Estimation of the accuracy of the transistor equivalent circuit using the input impedance First, we will consider the model amplifier (Fig. 7) with L = C = 0, i.e., with the unmatched impedances at the input. In this case, the input impedance of the amplifier can be assigned to the transistor. The frequency of the electromotive force of the input RF generator is given in the first column of Table 1. The amplitude of the electromotive force is set such that the amplitude of id(t) in the periodic oscillation mode is close to 80 mA for each of the four frequencies. The corresponding values of the maximum power of the RF generator are listed in the second column. The values of the microwave parameters found by the time-domain simulation (without using the IFET equivalent circuit) are given in rows ‘‘QHM only’’. The values of these parameters
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121
determined by the frequency-domain simulation (using the IFET equivalent circuit at sgs = 0.794 ps and sch = 1.283 ps) are given in rows ‘‘QHM & EC’’. Since the determination of parameters sgs and sch is a labor-intensive procedure, the results obtained at sgs = sch = 0 are also given in Table 1 (rows ‘‘QHM & EC0’’). The last column shows errors in the input impedance calculated with the IFET equivalent circuit. The impedance listed in rows ‘‘QHM only’’ is viewed as a reference because the 2D model (which is the base of the time-domain simulation) is higher than the equivalent circuit in the IFET model hierarchy.
Appendix A A.1. Derivation of expression (17) for the current in the input branch of the IFET equivalent circuit Consider ‘‘central’’ point c of the input branch of the IFET equivalent circuit in Fig. 1. Here, uc is the electric potential at this point, ugc = ug + ub uc, and ucs = uc us. Since ucs = Rgsiin, where iin is the current in the input branch, the equation ugs(t) = ugc(t) + ucs(t) gives
cgs ugs ¼ cgs ugc þ sgs iin ; 6. Estimation of the accuracy of the transistor equivalent circuit using the transducer power gain
where cgs(t) = Cgs[ugs(t), uds(t)]; sgs = Rgscgs is the time constant of the resistor–capacitor network (Rgs is a function of t). Differentiating both sides of this equation with respect to time, we have
The first two columns in Table 2 are the same as in Table 1. The third and fourth columns show the values of L and C providing the complex conjugate matching of the 50-ohm impedance of the RF generator with the transistor input impedance given in rows ‘‘QHM only’’ of Table 1. As above, rows ‘‘QHM & EC’’ were obtained at sgs = 0.794 ps and sch = 1.283 ps and rows ‘‘QHM & EC0’’ correspond to the assumption sgs = sch = 0. In Figs. 4 and 5, dashed and dash-dotted lines show cycles calculated at a frequency of 90 GHz, L = 0.0341 nH, C = 0.125 pF, sgs = 0.794 ps, and sch = 1.283 ps. The last column shows errors in the transducer power gain calculated using the IFET equivalent circuit. Gain values listed in rows ‘‘QHM only’’ are considered as reference ones.
dcgs dugs dcgs dugc diin ¼ þ sgs ugs þ cgs ugc þ cgs dt dt dt dt dt or
dcgs dugs dugc diin ¼ cgs þ sgs : ucs þ cgs dt dt dt dt Since
dcgs dcgs Rgs dcgs 1 iin ¼ sgs iin ucs ¼ sgs dt dt sgs dt cgs and
cgs 7. Conclusion According to our results, the use of the IFET large-signal equivalent circuit (with parameters sgs = 0.794 ps and sch = 1.283 ps) instead of the 2D model at frequencies of up to 60 GHz introduces an error not exceeding 5% for the input impedance (Table 1) and 0.6 dB for the transducer power gain (Table 2). The computer time expenses are as follows. For calculating the values of each row ‘‘QHM only’’ (in Tables 1 and 2), four oscillation periods were simulated. The calculation of the content of all eight rows ‘‘QHM only’’ took about 20 h of the 3-GHz processor, Intel Core 2 Quad CPU Q9650. For calculating characteristics Id(Ugs, Uds) and Qg(Ugs, Uds), the initial-boundary value problem for (1)–(6) was solved at 11 17 grid nodes (Fig. 3). It took about 24 h of the abovementioned processor. For calculating the values of rows ‘‘QHM & EC’’ and ‘‘QHM & EC0’’, the HIFETA code was executed 16 times. It took less than 4 s. This article should be considered as a technique for estimating the accuracy of the microwave FET equivalent circuit in comparison with the 2D model, which is higher than the equivalent circuit in the IFET model hierarchy. We use this technique for verifying the equivalent circuit of real transistors. To this end, the model amplifier (Fig. 7) is complemented by the substrate conductance (between points d and s) and output matching network. The HIFETA program (which is efficient in terms of CPU time saving) is used for optimizing power amplifiers, in particular, based on the boron-doped diamond structure similar to the depicted in Fig. 2.
ð22Þ
dugc ¼ iin ; dt
Eq. (22) takes the form
sgs
diin dcgs 1 dugs iin ¼ cgs þ 1 sgs : dt dt cgs dt
ð23Þ
Under the assumption that sgs is small compared with oscillation period T, we represent a solution to (23) in the form [19]
iin ðtÞ ¼ w0 ðtÞ þ
sgs T
w1 ðtÞ;
ð24Þ
where sgs/T is a small quantity. Substituting (24) into (23) yields the following expressions for terms of the zero and first orders of smallness:
w0 ðtÞ ¼ cgs
sgs T
dugs ; dt
w1 ðtÞ ¼ sgs
dcgs w0 dw0 sgs : dt cgs dt
Thus, (24) gives expression (17) for the current in the input branch of the IFET equivalent circuit. References [1] Kasu M, Ueda K, Kageshima H, Yamauchi Y. RF equivalent-circuit analysis of ptype diamond field-effect transistors with hydrogen surface termination. IEICE Trans Electron July 2008;E91-C:1042–9. [2] Chung JW, Hoke WE, Chumbes EM, Palacios T. AlGaN/GaN HEMT with 300-GHz fmax. IEEE Electron Dev Lett March 2010;EDL-31:195–7. [3] Radisic V, Deal WR, Leong KMKH, Mei XB, Yoshida W, Liu P-H, et al. A 10-mW submillimeter-wave solid-state power-amplifier module. IEEE Trans Microwave Theory Tech July 2010;MTT-58:1903–9. [4] Garber GZ. Analysis of nonlinear microwave circuits based on Schottky-gate field effect transistors (in Russian). Electron Eng Ser 2 1984;5:27–30. [5] Garber GZ. Numerical modeling of the characteristics of nonlinear equivalent circuits for s.h.f. Schottky-gate GaAs field-effect transistors. Soviet Microelectron March 1991;22:202–7. [6] Garber GZ. Method for calculating a small-signal equivalent circuit of extremely high frequency heterostructural field-effect transistors. J Commun Technol Electron July 2005;50:822–5.
G.Z. Garber / Solid-State Electronics 103 (2015) 115–121 [7] Ahlberg JH, Nilson EN, Walsh JL. The Theory of Splines and Their Applications. New York and London: Academic Press; 1967 [ch. 2]. [8] Stratton R. Diffusion of hot and cold electrons in semiconductor barriers. Phys Rev June 1962;126:2002–14. [9] Garber GZ. Modeling of operation of field effect transistors with submicron Schottky gate on GaAs (in Russian). Electron Eng Ser 2 1985;2:103–7. [10] Garber GZ. Quasi-hydrodynamic modeling of heterostructure field-effect transistors. J Commun Technol Electron Jan 2003;48:114–7. [11] Garber GZ. Model for simulation of AlGaAs–GaAs power heterostructure FETs. In: Proc IEEE Int Conf EUROCON, Belgrade; 2005. p. 867–70. [12] Reggiani L, Bosi S, Canali C, Nava F, Kozlov SF. Hole-drift velocity in natural diamond. Phys Rev B March 1981;23:3050–7. [13] Volpe P-N, Pernot J, Muret P, Omnes F. High hole mobility in boron doped diamond for power device applications. Appl Phys Lett March 2009;94:092102-1–2-3. http://dx.doi.org/10.1063/1.3086397.
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[14] Garber GZ, Dorofeev AA, Zubkov AM, Ivanov KA, Kolkovsky YuV. Design of the SHF power FETs on diamond base with assistance of computer simulation (in Russian). Electron Eng Ser 2 2013;1:9–16. [15] Samarskii AA. The Theory of Difference Schemes. New York: Marcel Dekker; 2001 [ch. 9,10]. [16] Scharfetter DL, Gummel HK. Large-signal analysis of a silicon Read diode oscillator. IEEE Trans Electron Dev Jan 1969;ED-16:64–77. [17] Powell MJD. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput J 1964;7:155–62. http://comjnl.oxfordjournals.org/content/7/2/155.full.pdf+html. [18] Hamming RW. Numerical Methods for Scientists and Engineers. New York: McGraw-Hill; 1962 [ch. 6]. [19] Cole JD. Perturbation Methods in Applied Mathematics. Waltham, MA: Blaisdell; 1968 [ch. 2].
Contents lists available at ScienceDirect
Solid-State Electronics journal homepage: www.elsevier.com/locate/sse
Estimation of the accuracy of the microwave FET equivalent circuit using the quasi-hydrodynamic model of electron transport Gennadiy Z. Garber Pulsar R&D Manufacturing Company, Moscow 105187, Russia
a r t i c l e
i n f o
Article history: Received 7 April 2014 Received in revised form 6 August 2014 Accepted 15 August 2014 Available online 11 September 2014 The review of this paper was arranged by Prof. S. Cristoloveanu Keywords: Equivalent circuit FET Frequency-domain simulation Quasi-hydrodynamic model Spline characteristics Time-domain simulation
a b s t r a c t Computational methods are considered for the time- and frequency-domain simulations of a microwave power amplifier with a transistor built on a diamond-like semiconductor. The first method is based only on a two-dimensional model of the intrinsic transistor that includes the quasi-hydrodynamic model of electron transport and electric field equations. The second method, which is less accurate but much more computer-time-saving, is based on the intrinsic transistor large-signal lumped-element equivalent circuit model with the parameters and spline characteristics calculated by means of the abovementioned two-dimensional model. Using these two methods, the microwave parameters of the amplifier are calculated for frequencies varying from 15 to 90 GHz. When using the equivalent circuit at frequencies of up to 60 GHz, the calculation error (equal to the difference between the results of the time- and frequency-domain simulations) does not exceed 5% for the input impedance and 0.6 dB for the transducer power gain. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Currently, field effect transistors (FETs) are widely used as the active elements of power amplifiers in the frequency range from several hundred megahertz to several hundred gigahertz [1–3]. Therefore, the adequacy of the intrinsic FET (IFET) large-signal lumped-element equivalent circuit model (Fig. 1) used for designing these amplifiers is becoming a key issue. In [4–6], we developed a method for calculating the characteristics and parameters of the IFET equivalent circuit that is based on the cubic spline construction [7] and the twodimensional (2D) quasi-hydrodynamic model of electron transport [8–11]. Below, we will calculate a number of microwave parameters of a power amplifier based on a transistor that ‘‘is made’’ on an n-type semiconductor, in which the field dependence of the drift velocity of electrons is close to the field dependence of the drift velocity of holes in boron-doped diamond [1,12–14]. The physical parameters of the semiconductor and insulator (Fig. 2) are close to the corresponding constants for diamond.
The microwave parameters will be calculated in two ways: using only the 2D model of the IFET that includes the quasihydrodynamic model of electron transport and electric field equations and combining the 2D model with the IFET equivalent circuit. 2. 2D model of the intrinsic transistor Under the operating conditions of the transistor, we can neglect the impact ionization of semiconductor molecules by electrons and holes. Then, the 2D quasi-hydrodynamic model of electron transport takes the following form:
@n ¼ r~jn ; @t ~jn ¼ l ! n E n þ rðDn nÞ;
http://dx.doi.org/10.1016/j.sse.2014.08.008 0038-1101/Ó 2014 Elsevier Ltd. All rights reserved.
ð2Þ
! @ðnuÞ u u0 ¼ r~jnu þ q~jn E n ; @t sn ~jnu ¼ n l ! E nu þ rðn Dn nuÞ; n
E-mail address: [email protected] URL: http://www.gzgarber.narod.ru
ð1Þ
n
n
ð3Þ ð4Þ
where u(x, y, t) = 3kTe/2 is the mean kinetic energy of electrons; x and y are the horizontal and vertical coordinates (Fig. 2); t is the time; k is the Boltzmann constant; Te(x, y, t) is the electron gas
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121
Fig. 2. The active region of the delta-doped MISFET on the diamond-like semiconductor. Fig. 1. Large-signal equivalent circuit of the IFET without impact ionization of semiconductor molecules by electrons and holes: ich is the channel current generator; Cgs and Rgs, the input capacitance and resistance; and Cgd, the transfer (feedback) capacitance.
temperature; r ¼
@ @ ; @x @y
The field dependence of the electron drift velocity appearing in (7) is given by
V n ¼ v e ½1 expðle Eus =v e Þ; 3
is the gradient operator; u0 = 3kT0/2 is the
mean kinetic energy of electrons in thermodynamic equilibrium; T0 is the lattice temperature (in our case, T0 is set equal to 300 K); n(x, y, t), ln(x, y, u), Dn(x, y, u) = (kTe/q)ln, sn(x, y, u), and nn(x, y, u) are the concentration, mobility, diffusion coefficient, energy relaxation time, and relative Peltier coefficient of electrons, jn ðx; y; tÞ and respectively; q is the positive elementary charge; ~ jnu ðx; y; tÞ are the electron flux density and electron energy flux ~ ! ! density, respectively; E ðx; y; tÞ is the electric field intensity; ln E ! is the electron drift velocity; and E ¼ j E j. ! Electric field intensity E ðx; y; tÞ is calculated using the electric field equations
!
rðe E Þ ¼ Q ;
ð5Þ
! E ¼ ru;
ð6Þ
where u(x, y, t) is the electric potential and e(x, y) is the absolute permittivity of the semiconductor or insulator; in our case, the relative permittivity is equal to 5.7 for both the semiconductor and insulator. Charge density Q is equal to zero in the insulator and to q(Nd n) in the semiconductor, where Nd(x, y) is the concentration of ionized donors. The kinetic coefficients for electrons are calculated by the formulas
ln ðx; y; uÞ ¼ V n ½x; y; Fðx; y; uÞ=Fðx; y; uÞ;
ð7Þ
sn ðx; y; uÞ ¼ T n ½x; y; Fðx; y; uÞ;
ð8Þ
nn ðx; y; uÞ ¼ K n ½x; y; Fðx; y; uÞ;
ð9Þ
where Vn(x, y, Eus), Tn(x, y, Eus), and Kn(x, y, Eus) are the given field dependences of the drift velocity, energy relaxation time, and relative Peltier coefficient, respectively, and Eus is the intensity of the uniform stationary electric field. When finding the field dependences, we assume that the semiconductor is homogeneous and its chemical composition is the same as at a point with coordinates x and y (in the semiconducting part of the transistor active region depicted in Fig. 2). Quantity Vn(x, y, Eus) is the drift velocity of electrons in such a semiconductor subjected to uniform stationary electric field Eus. Quantities Tn(x, y, Eus) and Kn(x, y, Eus) have the same meaning. Function F(x, y, u) involved in (7)–(9) is zero at u 6 u0 and is a solution to the energy balance equation,
qFV n ðx; y; FÞT n ðx; y; FÞ ¼ u u0 ;
ð10Þ
at u > u0. This equation follows from (2) and (3) at r = o/ot = 0.
2
ð11Þ 15
3
2
where le = 2 10 cm /(V s) at Nd = 10 cm and 500 cm /(V s) at Nd = 1018 cm3 and ve = 1.1 107 cm/s. The field dependence of the energy relaxation time in (8) has the following form: function Tn(x, y, Eus) is equal to vese/Vn(x, y, Eus) for Eus > Ee and to vese/ Vn(x, y, Ee) for Eus 6 Ee, where se = 1 ps and Ee = 3ve/le. With function Tn(x, y, Eus) taken in such a form, the abovementioned solution to (10) at u > u0 is unique. The field dependence of the relative Peltier coefficient in (9) is given by
Kn ¼
2 expðEus =Ee Þ þ 1: 3
System (1)–(6) is considered in the transistor active region depicted in Fig. 2. Boundary conditions are the following: n = Nd and Te = T0 at the source and drain contacts; ! jn , ~ jnu , and E is equal to zero at the the normal component of ~ contact-free semiconductor parts of the boundary; ! the normal component of E equals zero at the contact-free insulator parts; the electric potentials of the source, gate, and drain contacts (us, ug, and ud, respectively) are given constants (when we calculate the characteristics and parameters of the IFET equivalent circuit) or are determined by the extrinsic electrical circuit (when we simulate the transistor amplifier without using the IFET equivalent circuit). At the insulator–semiconductor interface, we require that only the normal component of ~ jn and ~ jnu be zero (because the permittivity of the semiconductor and insulator is one and the same). In addition to the formulated boundary conditions, initial conditions are necessary. They are arbitrary or are determined by the 2D distributions of u, n, and Te, which result from the previous solution of (1)–(6). For solving the above initial-boundary value problem, we solve at each time step Eqs. (5) and (6) by the method of matrix elimination [15] and then Eqs. (1) and (2) and, concurrently, (3) and (4) using the locally one-dimensional scheme [15] with jnx, jny, jnux, and jnuy expressed as the electron current density in [16]. Then we calculate the source, gate, and drain currents:
is ¼
ZZ
source
ig ¼
ZZ
q jny þ e
@ Ey dxdz; @t
@ e Ey dxdz; @t gate
ð12Þ ð13Þ
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121
Fig. 3. Grid on the gate bias – drain bias plane.
id ¼
ZZ
@ q jny e Ey dxdz; @t drain
ð14Þ
where z is the third axis directed perpendicularly to the plane of Fig. 2. In our case, the gate width along the z axis is equal to 1 mm.
Fig. 4. Output current–voltage characteristics (continuous lines) and 90-GHz cycles calculated without using (dashed cycle) and using (dash-and-dot cycle) the IFET equivalent circuit.
3. Equivalent circuit of the intrinsic transistor Calculation of the characteristics and parameters of the equivalent circuit depicted in Fig. 1 starts with constructing a rectangular grid on the plane with axes Ugs = ug + ub us and Uds = ud us (Fig. 3), where ub = 5 V is the built-in potential of the structure depicted in Fig. 2. This grid is uniform in Ugs and nonuniform in Uds; Ugs,i and Uds,j are the nodal coordinates; i = 0, 1, ..., m; j = 0, 1, ..., n; Uds,0 = 0. The values of Ugs,0, Ugs,m, and Uds,n must be such that the grid entirely covers the operating region of the IFET on the Ugs–Uds plane. At each node of the grid, we solve the initial-boundary value problem for (1)–(6). The 2D distributions of the electric potential, electron concentration, and electron gas temperature at t = 1, u(x, y) = u(x, y, 1), n(x, y) = n(x, y, 1), and Te(x, y) = Te(x, y, 1), respectively, are filed. This file also contains the time dependences of the following physical characteristics: the source, gate, and drain currents – is(t), ig(t), and id(t); the electrostatic induction vector flux to the gate – qg(t) = RR gate e Ey dxdz (it has the dimension of charge). The dynamic transconductance and dynamic conductance describing the channel current generator (ich in Fig. 1) are calculated as
SðxÞ ¼
GðxÞ ¼
sinðxsÞ
xs sinðxsÞ
xs
expðjxsÞS0 ;
ð15Þ
expðjxsÞG0 ;
ð16Þ
where S0 = oId/oUgs and G0 = oId/oUds are the static transconductance and static conductance; Id(Ugs, Uds) is the dependence of id(1) = Id on Ugs and Uds; x is the circular frequency; j is the imaginary unit; s = sch/2; and sch is the time of electron transit through the channel’s nonohmic part. Note that Id = Is = is(1) since the gate current at t = 1 is absent. Formulas (15) and (16) taking into account the electron transit through the channel’s nonohmic part correspond to the following time-domain expression for the channel current:
ich ðtÞ ¼
1
sch
Z
t
0
Id ½ugs ðt 0 Þ; uds ðt0 Þdt ;
tsch
where ugs(t) = ug(t) + ub us(t) and uds(t) = ud(t) us(t).
Fig. 5. Charge–voltage characteristics (continuous lines) and 90-GHz cycles calculated without using (dashed cycle) and using (dash-and-dot cycle) the IFET equivalent circuit.
The dependence of qg(1) = Qg on Ugs and Uds written as Qg(Ugs, Uds) determines the input and transfer capacitances according to the formulas Cgs(Ugs, Uds) = oQg/oUgs oQg/oUds and Cgd(Ugs, Uds) = oQg/oUds, respectively. The resistance in the input branch is given by Rgs = sgs/Cgs, where sgs is the time constant of the resistor–capacitor network. The current in this branch is calculated by the formula based on the assumption that sgs is small in comparison with the oscillation period:
iin ðtÞ ¼ w0 ðtÞ þ sgs
dcgs w0 dw0 : dt cgs dt
ð17Þ
Here, cgs(t) = Cgs[ugs(t), uds(t)] and w0(t) = cgs(t) dugs/dt. The derivation of formula (17) is given in Appendix A. The current in the transfer branch is given by
itr ðtÞ ¼ cgd ðtÞ
dugd ; dt
where cgd(t) = Cgd[ugs(t), uds(t)] and ugd(t) = ug(t) + ub ud(t). We use the cubic spline construction [7] to analytically continue the discrete dependences found by solving (1)–(6) at the grid nodes on the Ugs Uds plane depicted in Fig. 3. Therefore, the dependences Id(Ugs, Uds) and Qg(Ugs, Uds) may be called the spline characteristics of the IFET equivalent circuit.
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121
are required parameters sgs and sch for the node (4.5; 10 V): sgs = 0.788 ps and sch = 1.229 ps. They are determined using the TIFET macro for Microsoft Office Excel. 4. Simulation of the amplifier in the time and frequency domains In the model amplifier depicted in Fig. 7, the operating point of the IFET has the coordinates Ugs = 4.63 V and Uds = 10.06 V, at which the transfer capacitance is equal to Cgd = 0.014 pF. For comparison, the corresponding value of the input capacitance is Cgs = 0.347 pF, i.e., the assumption that Cgd = 0 used to determine the time constants, sgs and sch, is valid. After using the TIFET macro for calculating sgs and sch in the nodes (5; 10 V), (5; 12 V), and (4.5; 12 V), which (as well as the node (4.5; 10 V) considered above) are the closest to the operating point, we obtain sgs = 0.794 ps and sch = 1.283 ps for the IFET operating point (4.63; 10.06 V) with linear interpolation. These values of sgs and sch are the parameters of the IFET equivalent circuit. We will simulate the amplifier (Fig. 7) in the time and frequency domains for a given amplitude and frequency of the electromotive force of the input RF generator. In what follows,
Fig. 6. Time dependences of the (a) gate and (b) drain currents (continuous lines) and their exponential approximations (dashed lines).
A method for calculating parameters sgs and sch of the equivalent circuit will be considered below. We developed a version of the DIHEMT program [10] to solve the initial-boundary value problem for (1)–(6) at the grid nodes. The continuous lines in Figs. 4 and 5 show the calculated characteristics Id(Ugs, Uds) and Qg(Ugs, Uds), respectively. One execution of the DIHEMT code makes it possible to simulate all nodes of one vertical line on the Ugs Uds plane (Fig. 3): i = 0, 1, ..., m; j = const. Under the assumption that Cgd = 0, parameters sgs and sch can be found from the time dependences of gate current ig(t) and drain current id(t) as follows. In Figs. 6a and b, continuous lines are the ig(t) and id(t) dependences calculated in going from the node (5; 10 V) to the adjacent node (4.5; 10 V) and dashed lines (calculated by the least-squares method) represent the exponential approximations of ig(t) and id(t). The time constants of these exponential curves are equal to 0.788 and 1.229 ps, respectively. According to [6], these values
Ld(ud, ug, us), Lg(ud, ug, us), and Ls(ud, ug, us) are, respectively, currents at points d, g, and s determined by a linear active three port (LATP), part of the circuit in Fig. 7 that is outside the IFET unit; Md(ud us, ug us), Mg(ud us, ug us), and Ms(ud us, ug us) = Md + Mg are currents at points d, g, and s determined by the IFET unit. When simulating the amplifier in the time domain, the solution of the initial-boundary value problem for (1)–(6) lasts until periodic oscillations (with period T) set in. At each time step, the quadratic discrepancy
Fðud ; ug ; us Þ ¼ ½Md ðud us ; ug us Þ Ld ðud ; ug ; us Þ2 þ ½Mg ðud us ; ug us Þ Lg ðud ; ug ; us Þ2 þ ½Ms ðud us ; ug us Þ Ls ðud ; ug ; us Þ2
ð18Þ
of the IFET drain, gate, and source currents is minimized, with Ms (ud us, ug us), Mg(ud us, ug us), and Md(ud us, ug us) calculated by (12)–(14). The electric potentials at points d, g, and s (relative to ground, Fig. 7) are adjustable parameters. We use the Powell method [17] to minimize (nearly to zero) function (18) at each time step. Note that not only functions Ld
Fig. 7. Electrical circuit of the model amplifier: resistors of 0.3 and 2 X are the parasitic elements of the transistor.
119
G.Z. Garber / Solid-State Electronics 103 (2015) 115–121 Table 1 Calculation of the transistor input impedance. Frequency 1/T (GHz)
Power available from the RF generator (mW)
Simulation approach
Input voltage standing wave ratio
Real part of the input impedance (O)
Imaginary part of the input impedance (O)
Accuracy of input impedance calculation (%)
15
4.9
QHM only QHM & EC QHM & EC0
19.2 17.45 29.34
3.436 3.864 2.301
28.2 29.44 29.57
4.6 6.3
QHM only QHM & EC QHM & EC0
15.12 14.29 24.04
3.578 3.794 2.261
14.28 14.49 14.75
2 9.5
QHM only QHM & EC QHM & EC0
13.48 13.76 22.74
3.784 3.702 2.247
7.078 6.872 7.384
2.8 19.5
QHM only QHM & EC QHM & EC0
12.65 14.07 22.47
3.988 3.579 2.247
4.684 4.225 4.932
10 28.6
30
60
90
15.6
62.5
160
Table 2 Calculation of the transducer power gain of the amplifier. Frequency 1/T (GHz)
Power available from the RF generator (mW)
L (nH)
C (pF)
Simulation approach
Input voltage standing wave ratio
Output RF power (mW)
Transducer power gain (dB)
Accuracy of transducer power gain calculation (dB)
15
4.9
0.484
0.693
QHM only QHM & EC QHM & EC0
1.202 1.313 2.339
658.3 673.5 757.4
21.28 21.38 21.89
0.1 0.61
QHM only QHM & EC QHM & EC0
1.065 1.066 1.578
555 569.4 729.4
15.5 15.62 16.69
0.12 1.19
QHM only QHM & EC QHM & EC0
1.079 1.051 1.636
498.2 568.7 718.6
9.015 9.59 10.61
0.57 1.6
QHM only QHM & EC QHM & EC0
1.158 1.145 1.736
464.4 631.3 729
4.628 5.961 6.586
1.33 1.96
30
60
90
15.6
62.5
160
0.156
0.0572
0.0341
0.379
0.191
0.125
(ud, ug, us), Lg(ud, ug, us), and Ls(ud, ug, us) but also functions Md (ud us, ug us), Mg(ud us, ug us), and Ms(ud us, ug us) are linear. This assertion follows from the fact that, during minimization, we solve (5) with charge density Q taken from the previous time step. Since Ld(ud, ug, us), Lg(ud, ug, us), Ls(ud, ug, us), Md (ud us, ug us), Mg(ud us, ug us), and Ms(ud us, ug us) are linear, function (18) is quadratic, which facilitates the search for its minimum point. We developed a version of the DIHEMT program [10] for the time-domain simulation of the amplifier. The frequency-domain simulation, which is much more efficient in terms of CPU time saving, is based on the IFET equivalent circuit depicted in Fig. 1. We use the harmonic balance method (in our case, for five harmonics of the input RF generator frequency, 1/T) to obtain the periodic oscillation mode. To this end, the system of algebraic equations
M d ðud us ; ug us Þ ¼ Ld ðud ; ug ; us Þ;
ð19Þ
M g ðud us ; ug us Þ ¼ Lg ðud ; ug ; us Þ;
ð20Þ
M s ðud us ; ug us Þ ¼ Ls ðud ; ug ; us Þ
ð21Þ
is solved, where Md, Mg, Ms, Ld, Lg, and Ls are the complex amplitudes of currents at points d, g, and s determined by the IFET and LATP and ud, ug, and us are the complex amplitudes of potentials at points d, g, and s. Here, functions Md, Mg, and Ms are nonlinear. The system of nonlinear algebraic Eqs. (19)–(21) is solved by the Newton method with the Jacobian matrix including the derivatives of the cubic splines that describe the characteristics
of the IFET equivalent circuit. When analyzing this equivalent circuit, the direct and inverse discrete Fourier transformations [18] are performed. The Visual C++ program HIFETA (Harmonics In FET Amplifier) implements the frequency-domain simulation of the model amplifier. The calculated periodic time dependences of the currents and potentials (in both the time and frequency domains) govern the microwave parameters of the amplifier, some of which are given in Tables 1 and 2. Below, QHM stands for 2D model (1)–(6) and EC stands for the IFET equivalent circuit. The simulation frequencies are 1/T = 15, 30, 60, and 90 GHz. 5. Estimation of the accuracy of the transistor equivalent circuit using the input impedance First, we will consider the model amplifier (Fig. 7) with L = C = 0, i.e., with the unmatched impedances at the input. In this case, the input impedance of the amplifier can be assigned to the transistor. The frequency of the electromotive force of the input RF generator is given in the first column of Table 1. The amplitude of the electromotive force is set such that the amplitude of id(t) in the periodic oscillation mode is close to 80 mA for each of the four frequencies. The corresponding values of the maximum power of the RF generator are listed in the second column. The values of the microwave parameters found by the time-domain simulation (without using the IFET equivalent circuit) are given in rows ‘‘QHM only’’. The values of these parameters
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G.Z. Garber / Solid-State Electronics 103 (2015) 115–121
determined by the frequency-domain simulation (using the IFET equivalent circuit at sgs = 0.794 ps and sch = 1.283 ps) are given in rows ‘‘QHM & EC’’. Since the determination of parameters sgs and sch is a labor-intensive procedure, the results obtained at sgs = sch = 0 are also given in Table 1 (rows ‘‘QHM & EC0’’). The last column shows errors in the input impedance calculated with the IFET equivalent circuit. The impedance listed in rows ‘‘QHM only’’ is viewed as a reference because the 2D model (which is the base of the time-domain simulation) is higher than the equivalent circuit in the IFET model hierarchy.
Appendix A A.1. Derivation of expression (17) for the current in the input branch of the IFET equivalent circuit Consider ‘‘central’’ point c of the input branch of the IFET equivalent circuit in Fig. 1. Here, uc is the electric potential at this point, ugc = ug + ub uc, and ucs = uc us. Since ucs = Rgsiin, where iin is the current in the input branch, the equation ugs(t) = ugc(t) + ucs(t) gives
cgs ugs ¼ cgs ugc þ sgs iin ; 6. Estimation of the accuracy of the transistor equivalent circuit using the transducer power gain
where cgs(t) = Cgs[ugs(t), uds(t)]; sgs = Rgscgs is the time constant of the resistor–capacitor network (Rgs is a function of t). Differentiating both sides of this equation with respect to time, we have
The first two columns in Table 2 are the same as in Table 1. The third and fourth columns show the values of L and C providing the complex conjugate matching of the 50-ohm impedance of the RF generator with the transistor input impedance given in rows ‘‘QHM only’’ of Table 1. As above, rows ‘‘QHM & EC’’ were obtained at sgs = 0.794 ps and sch = 1.283 ps and rows ‘‘QHM & EC0’’ correspond to the assumption sgs = sch = 0. In Figs. 4 and 5, dashed and dash-dotted lines show cycles calculated at a frequency of 90 GHz, L = 0.0341 nH, C = 0.125 pF, sgs = 0.794 ps, and sch = 1.283 ps. The last column shows errors in the transducer power gain calculated using the IFET equivalent circuit. Gain values listed in rows ‘‘QHM only’’ are considered as reference ones.
dcgs dugs dcgs dugc diin ¼ þ sgs ugs þ cgs ugc þ cgs dt dt dt dt dt or
dcgs dugs dugc diin ¼ cgs þ sgs : ucs þ cgs dt dt dt dt Since
dcgs dcgs Rgs dcgs 1 iin ¼ sgs iin ucs ¼ sgs dt dt sgs dt cgs and
cgs 7. Conclusion According to our results, the use of the IFET large-signal equivalent circuit (with parameters sgs = 0.794 ps and sch = 1.283 ps) instead of the 2D model at frequencies of up to 60 GHz introduces an error not exceeding 5% for the input impedance (Table 1) and 0.6 dB for the transducer power gain (Table 2). The computer time expenses are as follows. For calculating the values of each row ‘‘QHM only’’ (in Tables 1 and 2), four oscillation periods were simulated. The calculation of the content of all eight rows ‘‘QHM only’’ took about 20 h of the 3-GHz processor, Intel Core 2 Quad CPU Q9650. For calculating characteristics Id(Ugs, Uds) and Qg(Ugs, Uds), the initial-boundary value problem for (1)–(6) was solved at 11 17 grid nodes (Fig. 3). It took about 24 h of the abovementioned processor. For calculating the values of rows ‘‘QHM & EC’’ and ‘‘QHM & EC0’’, the HIFETA code was executed 16 times. It took less than 4 s. This article should be considered as a technique for estimating the accuracy of the microwave FET equivalent circuit in comparison with the 2D model, which is higher than the equivalent circuit in the IFET model hierarchy. We use this technique for verifying the equivalent circuit of real transistors. To this end, the model amplifier (Fig. 7) is complemented by the substrate conductance (between points d and s) and output matching network. The HIFETA program (which is efficient in terms of CPU time saving) is used for optimizing power amplifiers, in particular, based on the boron-doped diamond structure similar to the depicted in Fig. 2.
ð22Þ
dugc ¼ iin ; dt
Eq. (22) takes the form
sgs
diin dcgs 1 dugs iin ¼ cgs þ 1 sgs : dt dt cgs dt
ð23Þ
Under the assumption that sgs is small compared with oscillation period T, we represent a solution to (23) in the form [19]
iin ðtÞ ¼ w0 ðtÞ þ
sgs T
w1 ðtÞ;
ð24Þ
where sgs/T is a small quantity. Substituting (24) into (23) yields the following expressions for terms of the zero and first orders of smallness:
w0 ðtÞ ¼ cgs
sgs T
dugs ; dt
w1 ðtÞ ¼ sgs
dcgs w0 dw0 sgs : dt cgs dt
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