time-domain modeling of impatt oscillators

988 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 7, JULY 1989

Time- Domain Modeling of IMPATT Oscillators

Abstruct -The dynamics of an IMPA'IT oscillator is modeled in phase space as a set of first-order differential equations. The numerically simple model contains approximations for the nonlinear field dependence of the ionization coefficient and for the effects of diffusion and series resistance in the field free region. The parameters are then adapted to a realized oscillator. First simulations show that the model gives reasonable results.

NOMENCLATURE Cross-sectional area of the diode. Capacitance of the series resonance circuit. Element of drift zone averager. Geometrical capacitance of the avalanche region. Geometrical capacitance of the drift region. Electric field. Normalization frequency. DC bias current through the diode. Reverse saturation current of the diode. Avalanche current. Influenced current at drift zone boundaries. Current through the series resonance circuit. Bias current density. Inductivity of the series resonance circuit. Length of the depletion layer. Length of the avalanche region. Length of the drift region. Length of space charge region. Length of space charge region when the avalanche region is not punched through. Doping concentration in depletion layer. Quality factor of the series resonance circuit. Load and losses of the series resonance circuit. Element of the drift zone averager. Series resistance per length unit of the n-layer. Reference period length for normalization. DC bias voltage across avalanche region. Avalanche voltage required for depletion layer punchthrough. Avalanche voltage required for avalanche zone punchthrough. DC bias voltage across drift region. Avalanche zone voltage deviation from the operating point. Voltage across avalanche region (U, = Uao + U=).

Manuscript received September 11, 1987; revised May 28, 1988 and February 6, 1988. This paper was recommended by Associate Editor T. Matsumoto.

The authors are with the Lehrstuhl f i r Hochfrequenztechnik, Technis- che Universitat Munchen, D-8000 Munich, Germany.

IEEE Log Number 8927737.

Drift zone voltage deviation from the operating point. Ionization coefficient. Coefficients for the approximation of a( E ) . Coefficients for the approximation of / a dx. Derivative of the ionization coefficient with re- spect to U,. Dielectric constant. Avalanche region carrier transit time. Drift region carrier transit time. Avalanche resonance frequency.

I. INTRODUCTION ESPITE their noisiness, IMPATT (IMPact Avalanche D Transit Time) diodes are still the main solid state

sources for millimeter-wave frequencies. Since 1958, when Read proposed this device [l], many publications have treated the calculation of output power, efficiency, impedance and the noise spectrum close to the frequency of oscillation. Most treatments separate into two groups: One tries to get as close as possible to device physics obtaining the diode behavior by electron statistics [2] or by solving partial differential equations [3], [4]. Incorporating physical descriptions this approach is useful to demon- strate secondary effects and the microscopic mechanisms making up the diode behavior. But the numerical effort required for this treatment limits its use for the examina- tion of transients and stationary states in an oscillator circuit, at least in the case of a resonator with high quality factor.)

The second group derives approximations for diode operation and solves them in the frequency domain (e.g., [l], [5] ) . This approach reduces numerical effort considerably and facilitates a detailed modeling of the oscillator circuit, but requires assumptions on the dynamics, the operating frequencies and rather good initial guesses at the Fourier components when the system's nonlinearities are strong.

So an intermediate approach making simple approximations based on device physics and evaluating them in time domain might be a competitive compromise or an additional tool taking into account following requirements for oscillator design:

simulation of the active device together with the circuit (no assumed excitation); ability to treat nonlinear effects such as quasiperiod- icity or a continuous emission spectrum;

0098-4094/89/0700-0988$01.00 01989 IEEE

GOELLER AND KAERTNER: IMPA'M OSCILLATORS 989

facilitation of nonlinear noise analysis; tolerable numerical effort.

In our approach the IMPATT oscillator model is given as a set of n ordinary first-order differential equations which describe the time development of the system's n independent variables in phase space. (Similar approaches which did not contain secondary effects were proposed by Claassen and Harth [6] and by Gupta and Lomax [7] at a time when computers for an extensive simulation were not available.)

In this form the model can be immediately submitted to a nonlinear noise analysis such as [8], [9] to calculate the noise spectrum of the oscillator without the help of linear superpositions. In Section I1 we will propose a model consistent with this general description, which will be adapted to an existing device in Section 111. Section IV describes the first results of a simulation using the adapted

model contains the well-known division of the depletion layer (length f ) into avalanche region (length I,) and drift region (length I,) which holds true for a Read type p'n-i- n+ structure (or the complementary one). It is also a more or less useful approximation in the case of p+n-n+ (or n+p-p+) structures where I , is taken as the length of the region in which most of the ionization (conventionally 95 percent) takes place. The approximation becomes less ac- curate for higher frequencies because the Read assumption that the avalanche zone has negligible thickness looses its validity. Then the variation of I , with U, in large signal operation should not be neglected, either. Still this division (with a constant I,) is the most simple way to model the device. Its usefulness has to be controlled by checking the results.

The diode model will be explained in detail in the following.

model. A. Avalanche Region 11. IMPATT OSCILLATOR MODEL

The model should describe the main physical effects of IMPA'IT operation which are avalanche multiplication and camer drifting. At high frequencies (above 50 GHz) a number of secondary effects becomes important (see [lo], [Ill). On the other hand, a model as low dimensional as possible is needed to study the principles of IMPATT oscillator dynamics.

This simple model is shown in Fig. 1. The equations governing the system behavior are

d R 1 d t iw L ' L - = - 1 , - -(U, + U,+ u c )

d 1 -U, = - ( i w - i ) dt C,

d 1 -U, = - ( i w - i ,) dt C,

d 1 - U c = - 1 , dt C

(3)

(4)

d - i = [ ( C d x - 1) ( i + I , ) + I, dt r,

d 2 - i e = - ( i - i e ) . dt rd

(The voltage across the diode terminals is U,, + U, + + ud. Note that U,, + the dc bias in the operating point, does not enter into system dynamics except as a parameter in the ionization coefficient a. The state variables U, and U, may have an additional dc component due to rectifica- tion.)

Stable operation of the IMPATT oscillator is possible only near a series resonance of the resonator. Close to that frequency the resonator can simply be modeled by a series resonance circuit though in a refined model for better quantitative results it is important to match the impedance of the resonator over a wide frequency range. The diode

The geometrical capacitance of the avalanche region C, is given by

€ A c =-

a I ,

The avalanche dynamics is implemented into the diode model by the Read equation which makes the assumption of equal ionization rates and saturated velocities for both kinds of carriers. Its general form (5) contains /adx , which has to be replaced by an analytical expression. We sup- posed the ionization coefficient a to depend immediately on the electric field (this is a rough simplification for frequencies above 50 GHz as the mean free paths of the carriers cannot be neglected any more compared to the depletion zone length). This dependence is approximated by [I21

The parameters a1 and PI are determined from the dependence of the ionization coefficient of Si on the field strength [13, p. 1551. Using the Schottky approximation (totally depleted space charge region) and in the special case of a p+n-n+ structure (n-layer length f = I, + fd), the electric field as a function of the voltage across the avalanche zone U, is given as


i

Fig. 1. IMPATT oscillator model.

where

U,, = U,, + U,. Note that the length of the space charge region (I, resp.

(8), the first (punchthrough of the depletion layer) was valid during all times of the cycle in the simulations carried out. In a quantitative modeling, the influence of the space charge on the avalanche process which has been neglected in above equation should be accounted for.

Using (8) for integrating a over the n-layer we found out that 0

I,,) is time dependent with U,. Of the three cases given in 1

0 1 2 t 'd

Fig. 2. Response functions for different drift zone models. (a) No diffusion. (b) Exact solution [14]. (c) Approximation by a low-pass filter.

i Iadx = a 2 e - 8 2 / ( u a o f U , ) (9)

approximates dx well, i.e., withn ~ percent for the interesting range of U, and the parameters given in Section 111.

comes Thus the analytical form of the Read equation (5) be-

B. Drift Region

given by The geometrical capacitance of the drift region C, is

€ A c d = -

'd

The influenced current at the terminals of the drift zone is the average over the drifting charge weighted with the

velocities of the carriers. In the limit of zero diffusion and saturated carrier velocities this means simply averaging the avalanche current i over the drift zone transit time r d . The averaging can be modeled by a linear system with rectangular response function h ( t ) (Fig. 2(a)). If the efforts of diffusion in the drift zone are taken into account, the response function has to be modified to the one derived by Gannett and Chua [14] (Fig. 2(b)). This response function is an exact solution for the drift process with diffusion and is similar to the response function of a low-pass filter. Thus we approximated the exact solution by expanding the transfer function

(11) e -Jw'd ) H(o) = -(l- 1

J w rd

of an ideal averager with rectangular response function

GOELLER AND KAERTNER: IMPATI OSCILLATORS 991

and the transfer function of a low pass

into Taylor series around w?d = 0 and comparing the first and second terms of the two series. This yields

gR, = 1

(13) ‘ d C,R,= 2.

Fig. 2(c) shows the response function of this low-pass filter which is a good match for Gannett’s exact solution though it corresponds to a somewhat higher diffusion constant. The great advantage of modeling the diffusion by a low- pass filter is that it contributes only one additional degree of freedom to the system, i.e., only one additional differential equation. The exact solution as well as the ideal averager represents a distributed system having an infinite number of degrees of freedom.

The low pass describing the drift process is implemented as a current source i, (see Fig. 1 and (6)) which is controlled by the voltage U, across the capacitor of a low pass driven by the avalanche current I.

To compare our approximation with small signal results, we calculated the small signal impedance of the diode using the standard linear approximation

for the ionization rate, obtaining

w 1 +- w:c, 1 - ( w/w,)’

with

; WO=/-. e=- O l d

“S

The expression is almost identical to the one obtained when using Read’s and Gilden’s approximations [15], except for the nominators containing 8. The real part of 2, in (14) is about 40 percent lower than that given in [15], where diffusion is neglected. The effect of diffusion leads to a significant decrease in the negative resistance for frequencies above 20 GHz [lo].

If the reverse voltage across the diode is below punchthrough (Is < I ) during parts of the cycle, (l), (3), and (6) have to be modified:

R - . R + R , . ( l - l , )

1 s - ‘a I d

?d ?d * -

where R , is the series resistance per length unit of the n-layer, calculated to be 4770 Q/cm. Note that the terms replacing R, c d , and ?d are time dependent via I,.

111. ADAPTION OF THE MODEL TO A REALIZED OSCILLATOR

The model was adapted to an oscillator realized by Buchler et al. [16]. There were different diodes used; mentioned below are the parameters selected for simulation:

structure: p+n-n+ diode diameter: 40 pm length of n-layer: 375 nm doping concentration in n-layer: No = 1.5. 10’’ cm-3.

The ratio I d / / , was assumed to be 4. For this partition, about 90 percent of the ionization at breakdown takes place in the avalanche region.

Calculation of the electric field E(x) for the given doping concentration ND revealed that there would be a large part of the n-layer free of electric field at breakdown. This would mean that the mean speed of carriers is re- duced in the space charge region and that the field free rest of the depletion layer had to be modeled by a series resistance. Under these circumstances a higher bias current is needed in simulation to make these devices oscillate. This discrepancy between experiment and simulation probably stems from the fact that at these frequencies the assumption of a only depending on E ( x , t ) is no longer valid (i.e., the distance which the carriers travel between collisions is no longer negligible in comparison with drift zone length so the speed of the carriers in the drift region corresponds to a virtually higher electric field). Adding to that is the fact that ionization rates decrease with rising temperature. Because of this dependence, which is not contained in the model, breakdown voltage rises with temperature causing higher fields at breakdown than a calculation at room temperature predicts. These effects could be taken into account by introducing a correcting factor for the doping concentration N,, but in first simulations we simply lowered its value to 10l6 ~ r n - ~ to ensure saturated carrier velocities throughout the diode.

The multiplication factor, i.e., the ratio between bias current I,, and reverse saturation current I,, was assumed to be lo6 [lo].

The resonator is a disk resonator using highly insulating silicon as dielectric material [16]. An impedance plot (Fig. 3) calculated from Buechler’s formulas for the disk resonator [17] shows a series resonance at 90.72 GHz and a parallel one near 70 GHz. As the operating frequency of the oscillator should be close to 90 GHz, we restricted ourselves to one series resonance circuit at 90.72 GHz in the first simulation. The values of the elements were taken from the impedance plot close to the resonance. For a more exact simulation it is very important to improve the resonator model over a wide frequency range as oscillations far from the carrier are one important issue of nonlinear time-domain modeling. It will be shown though that even this simple model exhibits considerable activity in a wide frequency range (Hines already derived in [18] that in the presence of a strong signal parametric oscillations can occur even if the circuit is nearly open or shorted at these frequencies.)

992

, o o o ~ Resistance R, Reactance X / n

300‘

300-

-40 1 23 45

f0

I

+-p= X

1 .

1.1 88

f GHz

Fig. 3. Resonator impedance 2 = R + j X .

IV. SIMULATION For the simulation the frequency was normalized with

fo = 91.5 GHz, the frequency that n-layer length was opti- mized for, and the time with To = l/f,, so that one period of a limit cycle should take about unit time. The set of differential equations was integrated using a standard Runge-Kutta algorithm with 100 steps per unit time (in most cases). In general the solutions were very insensitive to variations of the step length. Only PoincarC and return maps required smaller steps.

As the system is rather small, we used simple forward integration for the determination of stationary states. Transient processes lasted for about 1000To while the equations were integrated for more than 5000 up to 800 OOOT, per parameter value. (Integrating over 5000T0 took about 2 h on an AT compatible or 15 min on a Microvax 11). If the system is higher dimensional or has long transients, the periodic stationary states have to be determined by a boundary value problem solver.

The dynamics of an autonomous system (a system without external excitation) is completely determined in the phase space spanned by the independent variables (state variables), e.g., the voltages across the independent capaci- ties and the currents through the independent inductivities. In our model the phase space is the six-dimensional Carte- sian space W6, with elements x’= (i,, U,, Ud, U,, i, i,). Peri- odic oscillation corresponds to a closed curve (trajectory) in phase space. There are different possibilities to visualize system behavior in a two-dimensional plane:

choosing two “characteristic” state variables (corresponding to an x-y-representation on the oscillo- scope); displaying one state variable over time; Poincart maps: two state variables are displayed when the trajectory intersects a (n - 1)-dimensional plane (PoincarC plane). This plane can be determined by a constant value of one state variable. (It is best

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 7, JULY 1989

- . 2 0 . 2 Projection of the limit cycle onto the (U, + ud, i,)-plane, ob- Fig. 4.

tained for J, = 10 kA/cd and Q = 550.

to choose the state variable showing the most regular behavior to determine the Poincart plane.); return maps: The value of one state variable at a Poincart section is displayed as a function of its value at the preceding PoincarC section.

The latter two methods are particularly useful to character- ize quasiperiodic and chaotic system behavior, provided that transients have died out. (For a introduction to quasi-periodic and chaotic systems, see, e.g., [19].)

A . Comparison with Experimental Results The oscillator realized by J. Buechler and J. F. Luy in

1986 was measured at a bias current I , =10 kA/cm2. The resonator had a quality factor Q = 550. The ratio of internal resonator losses to load was 2/1. In this case, an output power of 10.7 mW was obtained (private communi- cation).

Fig. 4 shows the projection of the trajectory in the stationary state onto the (U, + u d , i,)-plane obtained in simulation. The stationary state is a limit cycle with period 4 T,, i.e., there exist subharmonics at frequencies f0/4, f0/2, and 3f0/4. When we project of the trajectory unto the i,-axis, the sum of the power dissipated in the resonator and the power delivered to the load can be calculated from

whereA R is the resistance of the series resonance circuit and I , is the amplitude of the current through the resonator. I , is nearly sinusoidal due to high quality factor of ‘,he resonator. For the parameter values given above we get I , =120 mA; so the output power calculated from (17) is about 1.8 mW. The dependence of the output power on the

GOELLER AND KAERRJER: IMPA'TT OSCILLATORS

t 'load my

9 4

l o 4 - .

l o 3 -

102-

993

0 10 20 30

Fig. 5. Output power obtained in simulation as a function of bias current density, for Q = 550.

L

+ x X X

+ no o s c i l l a t i o n

l i m i t cyc le w i t h period To % b i f u r c a t e d l i m i t cyc le w i t h per iod "To

o quasiper iodic behaviour

0 0 0 0 Y X 4 3

x 0 0 0 % x x o 9 6 4 4

0 Y 2

I

5 I

10 I

15 1 -

20 Jo / kA/crn2

Fig. 6. Stationary states for different values of J, and Q.

bias current (see Fig. 5 ) has the form which is usually (e.g., [20]) obtained in IMPATT oscillator measurements (low output power-steep increase-saturation). The fact that the calculated values are considerably lower than the measured one is probably due to the oversimplification of the external circuit and to an excess modeling of the effect of diffusion. In future simulations the low pass R,C, of Fig. 1 should be replaced by a combination of a low pass and an all pass.

Note that a large-signal analysis in frequency domain (proposed by Harth [21], [22]) which neglects diffusion

predicts much more output power (42 mW) than the experiment yields.

B. Dependence of the Oscillator Dynamics from Bias Current and Quality Factor

We have calculated the stationary states of the oscillator circuit for several values of the bias current I , and quality factor Q of the resonator (diode parameters were held constant). The kind of stationary state for the different parameters is shown in Fig. 6. The general feature of this graph is that the oscillator dynamics tends to become more


- 3

-. 15 0 . 1 5

Fig. 7. Projection of the stationary state onto the (U, + U,,, i,)-plane for Jo = 5 kA/cn? and Q = 550.

complicated with higher quality factor. So while the output spectrum gets narrow with increasing Q, the oscillations within the diode spread over more frequencies. This in turn may influence the output via the noise measure or locking effects, which might be the case for frequency hopping during tuning.

One type of stationary state observed in simulation is a limit cycle with a periodicity of a multiple of To (see Fig. 4 for an example). The other kind is a stationary state showing no obvious periodicity. To classify this second state more precisely, we investigated the oscillation for a bias current of I o = 5 U/cm2 and a quality factor of Q = 550 more accurately. In Fig. 7 the projection of the phase space trajectory onto the (U, + ud, i,) plane is shown over an integration interval of 1000O-To after dying out of transients. The trajectory shaded a whole area in this plane. To get more information about the behavior we calculated the autocorrelation spectrum Ci, of the resonator current i, and Ci of the avalange current i shown in Fig. 8(a), (b) respectively. This is carried out by sampling the corresponding time series with sampling time of TA = T0/5, respectively To/lOO and subsequent fast fourier transformation. One can see that the spectrum is built up by the frequencies f i , f 2 and combination frequencies nfl + mfi with n, m E 2. Note that the two frequencies are related by

f l + f 2 = fo (18)

with f o the resonance frequency of the resonator. Compar- ing Fig. 8(a) and (b) one can see that the spectral lines aside from f o are strongly suppressed by the filter characteristic of the resonator. Like subharmonic generation this kind of behavior has also been experimentally observed by Rydberg and Lewin [23] and is known as parametric instability, see also [18] and [24]. The time behavior of

0

- 5 0

-100 0 100 f [acz] 200 0 250 f cm21 500

(a) (b) Fig. 8. Normalized autocorrelation spectra of (a) resonator current and

(b) Avalanche current for Jo = 5 kA/cn? and Q = 550.

. l o

.oo

- .071 , . . I . . . . I - . 3 - 0 5 UTo 10 0 5 '/To 10

( 4 ( 4 Fig. 9. Simulated time dependence of some state variables in the sta-

tionary state for Jo = 5 kA c d and Q = 550. a) Avalanche current. (b) Resonator current. (c) dr i f t current. (d) AV h anche voltage.

same state variables is shown in Fig. 9(a)-(d). Note that the resonator current i , is almost of sinusoidal shape whereas the internal variables of the diode, the avalanche current i and voltage U,, are strongly modulated. As i, is the only state variable accessible to measurement, the difference between the behavior of i, and the internal variables of the diode indicates that even a narrow output spectrum need not justify a harmonic analysis, as the impedance Z(o, A) may be considerably altered in the presence of parametric signals (see 171, [25]).

GOELLER AND KAERTNER: IMPATC OSCILLATORS 995

i w f A -

-0.1 fl 0.1

Projection of the Poincark map at i =10 mA (both directions Fig. 10. plotted) onto the (U, , , i,)-plane for J, = 5 k A / c d and Q = 550.

The correlation spectrum of the avalanche current, shown in Fig. 8(b) has all features of the correlation spectrum produced by a quasiperiodic flow caused by the two frequencies fi and f2 which have no rational relation- ship. Two frequencies related by

P fi = ;f2

( p , q integers) would produce a limit cycle with period

by combining (18) and (19). A quasi-periodic flow in contrast produces a two dimensional torus in the phase space as an invariant set of the flow, i.e., the torus is covered by the trajectory of the system in the course of time. This can be verified by calculating projections of PoincarC maps. In practice, of course, there is no difference between a quasiperiodic state and a periodic one characterized by very large integers p and q.

Fig. 10 shows the projection of a PoincarC map onto the (U,, i,)-plane generated by the points where the trajectory crosses the five-dimensional plane defined by i = 10 mA in both directions. The points shown denote a closed curve filled up in the course of time. This curve is an image of the cross section through the torus.

V. CONCLUSION The model proposed above allows the calculation of

IMPATT oscillator dynamics with comparably low numerical effort. The first simulations gave reasonable results for output power and dynamical behavior of IMPAIT devices. It has been shown that small variations in dc and circuit parameters cause not only quantitative but qualita- tive changes in diode behavior, e.g., transitions from periodic to quasi-periodic or chaotic oscillations. This ob- servation which cannot be modeled by the common fre-

T = ( P + 4)To

quency-domain approaches agrees well with experimental experience, such as frequency hopping or broad-band oscillations observed in tuning. The model’s capability of quantitative predictions has to be checked by measuring realized oscillators over a wide range of operating points and refining the basic model.

ACKNOWLEDGMENT The authors wish to thank Prof. Dr. Peter Russer and

Dipl. Ing. J. Buchler for many helpful discussions, as well as the reviewers for their constructive comments.

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1394-1398, Oct. 1984.

ED-13, pp. 169-175, Jan. 1966.

from the Tcchirical University of Alrrrk Germany, in .I!%.

nonlinear circuits.

time-domain modeling of impatt oscillators

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