64
IEEE
TRANSACTIONS
ON
ELECTRON
DEVICES,
VOL
ED-16,
NO.
1, JANUARY
1969
Large-Signal Analysis of a Silicon
Read Diode Oscillator
D.
L. SCHARFETTER,
MEMBER, IEEE, AND H. K. GUMMEL,
Abstract—This paper presents theoretical calculations of the
large-signal admittance and efficiency achievable in a silicon p-n-v-n
Read IMPATT diode. A simplified theory is employed to obtain a
starting design. This design is then modified to achieve higher
efficiency operation as specific device limitations are reached in
large-signal (computer) operation. Self-consistent numerical solutions are obtained for eguations describing carrier transport, carrier
generation, and space-charge balance. The solutions describe the
evolution in time of the diode and its associated resonant circuit.
Detailed solutions are presented of the hole and electron concentrations, electric field, and terminal current and voltage at various
points in time during a cycle of oscillation. Large-signal values of the
diode's negative conductance, susceptance, average voltage, and
power-generating efficiency are presented as a function of oscillation
amplitude for a fixed average current density. For the structure
studied, the largest microwave power-generating efficiency (18
percent at 9.6 GHz) has been obtained at a current density of 200
A/cm?, but efficiencies near 10 percent were obtained over a range of
current density from 100 to 1000 A/cm*.
ELF-CONSISTENT large-signal numerical solutions are obtained for equations which describe
carrier generation and space-charge balance in a
silicon p-n-v-n
Read
diode
[1] microwave oscillator.
The solutions describe the evolution in time of the diode
and
associated
circuit
(see Fig.
1). The
of the detailed
IEEE
calculation,
in addition
doping
profile
of the diode (see Fig. 2) was evolved starting with de-
to
the enhancement of our understanding of the oscillation
mechanisms, was to find conditions under which high
efficiency and power output can be obtained, and to
establish theoretical limits for these quantities. A
rigorous optimization, however, was not attempted in
this exploratory work. This would have had to start
with carefully formulated constraints and would have
constituted a very large-scale project. However, the
largest efficiency obtained in the detailed numerical
calculation reported here is 18 percent, in good agreement with the simplified theoretical estimate.
The large-signal operating characteristics of the
diode are calculated in the following two ways:
A.
Voltage Driven
To study the performance at a given freguency,
1. INTRODUCTION
S
The aim
MEMBER,
but
for various bias and load conditions, it is convenient to
drive the diode with a sinusoidal voltage applied
through a coupling capacitor [see Fig. 1(a)]. From such
studies, the admittance at the fundamental freguency
is obtained as a function of driving voltage amplitude.
Likewise, the ac power delivered by the diode into the
voltage generator is obtained.
sign considerations discussed in [2], and reviewed here,
B.
and modified after initial results for improvement of
efficiency. From the numerical solutions, the large-signal operating characteristics of the diode and the efficiency at the fundamental freguency (defined as ac
power delivered by the diode divided by dc power
dissipated) are computed.
bedded in a resonant circuit [see Fig. 1(b)] and performs as a free-running oscillator. In as much as the ac
voltage in the oscillator case is nearly sinusoidal, both
calculations should, and indeed do, give comparable
Read in his original paper [1] suggested that an efficiency of 30 percent should be obtainable in a silicon
diode. This calculation, however, neglected the finite
width of the avalanche region, and neglected the diff
ence in hole and electron ionization coefficients in si
con. A modification of Read's calculation, which includes the finite width of the avalanche region and the
experimental values of ionization coefficients for silicon
[3], is presented
in Section
II. It is found
that the
avalanche region must have an appreciable width for
the diode to have high power capability and high
efficiency. A theoretical efficiency of the order of 15
percent is obtained for conditions which should approach maximum power output.
Manuscript received June 3, 1968.
e suthors are with Bel Telephone Laboratories, Inc., Murray
i
Free-Running Oscillator
For another series of calculations,
the diode is im-
results.
In Section III, we present the large-signal operating
characteristics and detailed solutions. The operating
characteristics, i.e., admittance parameters, average
voltage, and power generating efficiency, are shown as
a function of diode ac voltage amplitude in Figs. 4
through 9. Fig. 10 is a diode admittance plot (small and
large signal) which shows the variation of conductance
and susceptance with oscillation amplitude and freguency. Also indicated are the power generating efficiencies. Fig. 10 is a compact presentation of the important results of this paper. Detailed solutions are pre-
sented of the hole and electron concentrations, electric
field, and terminal current and voltage at various points
in time during a cycle of oscillation. These solutions are
shown in Fig. 11 (Ja.—200 A/cm?, freguency
> 11.4
GHz)
and
Fig.
12
(Js.-1000
A/cm?,
freguency
=13.4 GHz). The dynamics of charge-pulse buildup
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SCHARFETTER
AND
GUMMEL:
SILICON
READ
DIODE
c
|c1€
OSCILLATOR
65
one-half the average voltage
drift region
(a)
c
it
€
L
$
€=
Diode and associated circuit. (a) Voltage driven. (b) Free-
running oscillator.
Vel
,
(Vaed + Vaea)
()
the total dc voltage is divided between
across
the avalanche
region
Va
the part
and
the
part developed across the drift region Vue.p. Note that
we are neglecting any appreciable average ac power contribution from the avalanche region; this is because the
avalanche region voltage is inductively reactive relative
d
%59h
to the particle current [4]. The displacement current is
|
Z
capacitively reactive relative
|
% 0"
to the diode
voltage and
therefore contributes no average ac power. The use of a
|
£
simple eguivalent circuit [4] for the avalanche region,
z
which is derived from small-signal considerations, may
of course have no validity under large-signal conditions.
â 10
However, we will use this approach as a starting point
£ o'
H
H5y
5
2
109
14
where
developed
1022
o
2
7
(b)
Fig. 1.
across the
half the inverse transit time of the drift region), the
motion of O, results under favorable conditions in an
ac particle current which is =180 degrees out of phase
with the ac voltage across the diode. The average of the
particle current is the dc current Ja.. The particle current swing is, therefore, at most from zero to twice the
dc current. For a sguare wave of particle current and a
sinusoidal variation of drift voltage, both with magnitude and phase as described above, the microwave
power-generating efficiency e is
D
o
T'u.p developed
[1], [2]. At the drift frequency (fp=one-
for a simple design theory, and compare detailed
merical calculations with the simple treatment.
In
S
4
5
6
DISTANCE IN MICRONS
7
s
o
in the avalanche region and drift through the drift
region is in gualitative agreement with Read's original
prediction of operation.
Section IV presents conclusions, and the Appendix
describes the time evolution computer program.
II. SIMPLIFIED DIODE DESIGN
In this section, we extend Read's efficiency calculation to include the dc voltage developed across the
avalanche region, and to allow for the unegual ioniza-
as appropriate
it was
assumed
that
no dc
produce an ac particle current amplitude egual to the
A versus distance.
electrons,
calculation,
power loss was associated with the avalanche region
since it was considered small in extent compared to the
drift region, and that such an avalanche region could
Fig. 2. Net impurity concentration of Read diode
tion rates of holes and
Read's
nu-
for
silicon [3]. As pointed out by Read, high-power opera-
dc current when operated at the drift freguency and
operated with ac voltage amplitude egual to one-half
the dc voltage [same assumptions as used above in
deriving (1)]. Since Vu.4 was neglected compared to
Vuep. Read predicted an efficiency of 1/m or about
30 percent.
Small-signal calculations [1], [2], [4], [5] show that
a resonance frequency f, is associated with any avalanche region, and that it increases as the sguare root
of dc current. To obtain small-signal negative resistance reguires operation above this freguency. How-
ever, operation of an avalanche region at a freguency
well above its resonance freguency was found to result
in very inefficient oscillations in a limited number of
exploratory large-signal calculations that were completed on a variety of Read diode structures. Small-sig-
tion reguires the generation of as large a charge pulse
nal calculations in the freguency domain [2] showed an
Quax as possible in the avalanche region, without a reduction of the electric field in the drift region below that
reguired for velocity saturation when this charge moves
optimum growth factor for small-signal oscillations occurring when the operating freguency was about 20
percent higher than the resonance freguency. Therefore,
we take as a design criterion that the avalanche freguency of a Read diode should be related to the drift
through the drift region. The
motion of Quax through
the drift region results in an ac voltage amplitude about
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66
IEEE
evaluated
frequency of the diode by the following condition:
fp
1.2fn.
2
The small-signal constraint above, coupled with largesignal conditions imposed on the driít space to be discussed below, results in a tentative design for the Read
diode which is analyzed numerically in this paper.
The sguare of the avalanche region resonance freguency is proportional to the dc current density Juc, the
carrier scattering limited velocity ./, and the partial
derivative of the average ionization rate By for electrons in the avalanche region with respect to the average
clectric field £4.
is [1], [2], [4]
An approximation for the relationship
(nfa):
= 2By'Jae/ e
G)
where e is the dielectric constant, and By’ the partial
derivation of B, with respect to Êx. The dc voltage
developed across the avalanche region is defined as
Va
= EaLa,
and the dc voltage developed
similarly defined as
S
across the drift region is
Va = EpLp,
where
TRANSACTIONS-ON
G)
drift regions.
With
the definition
of drift
fo — /2Ly,
(6)
Tuep can be expressed as
Vaen = Eptd/2fp.
For 50-percent
voltage
(7
modulation
("™~}
lucp),
Onax
from Gauss' theorem is about —eÊp/2. The ac particle
current
(which
for
100-percent
current
modulation
equals the dc current) eguals On,«fn; therefore, the dc
bias current density for efficient operation of the drift
space is proportional to freguency:
Ja = «Enfp/2.
®)
For ionization rates appropriate for silicon [3], we find,
for junctions with breakdown fields between 3 and
4X10*V/cm, that the integral of the electron ionization
rate with distance is not unity, as is the case for egual
rates, but about 3:
B,
(9)
3.
If we assume for simplicity that B, varies as the sixth
power of E,, we obtain
DEVICES,
JANUARY
conditions
given
(10),
(8), and
(Orfn)* m 2X 6 X 3 X Vaenfn*/Vaeas
()
for
the
by
1969
(7),is
and with the condition on fs and fp given by (2) results
in a relation between Ve and Vaep:
Va > 11Vap.
12
This result is only gualitatively correct, since it is based
on a combination of small-signal results for the avalanche region, and large-signal constraints on the drift
region. Furthermore, the factor 2 in (3) and (11), the
factor 6 in (10) and (11), and the factor 3 in (9) and
(11) are all approximate, and depend to some extent
upon dc bias, freguency, and oscillation amplitude.
However, the gualitative result is important: the avalanche region voltage, for a Read diode designed for
efficient high-power oscillator operation, is on the order
of the drift voltage and not negligible as suggested by
Read. This condition reduces the theoretical efficiency
by about a factor of 2 (15 percent instead of 30 percent).
Note that for Ge, GaAs, or other materials with nearly
egual ionization rates, the factor 3 in (9) becomes a 1
and Va3 Vaep, hence the theoretical efficiency is of
the order of 23 percent.
IlI. LARGE-SIGNAL CALCULATION
Ly and Ly are the effective widths of the av-
alanche and
freguency
ELECTRON
In this section, we discuss the precise large-signal
operating characteristics of a particular Read diode
oscillator, as obtained by a numerical calculation per-
formed on a high-speed digital computer. Details of the
computer program are discussed in the Appendix.
Briefly, the approach is to obtain self-consistent numerical solutions for the eguations describing carrier
transport, carrier generation, and space-charge balance
in a one-dimensional semiconductor structure. The
solutions describe the evolution in time of the diode and
its associated resonant circuit. Basically, the program
solves the following problem at various instances of
time during a cycle of oscillation. Given the instanta-
neous distribution of the hole and electron concentrations and terminal boundary conditions, how will the
carriers move in time, i.e., how will the system, diode,
and circuit time evolve? Given the carriers, the program
determines the electric field by Poisson's eguation, and
knowing the electric field and carrier concentrations, it
obtaims the instantaneous hole and electron particle
currents (including the dependence of carrier mobilities
upon impurity concentration
[7] and electric field
strength
generation
[8], [9]). From the particle currents and net
rate
(including
generation
and
tion by field-dependent impact ionization
recombina-
[3] and car-
rier concentration-dependent single-level recombination
centers [10], [11]), it obtains the time derivative of the
or, using (9) and (4),
By
Equation
6 X 3/Vaca-
(3), the avalanche
resonance
(10)
relation,
when
carrier concentrations from the continuity equations.
From the time derivatives of the carrier concentrations,
the program computes the carrier distributions an instant in time later, and repeats the cycle.
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SCHARFETTER
AND
GUMMEL:
SILICON
READ
DIODE
OSCILLATOR
The diode operating characteristics are calculated for
the diode imbedded in the circuit shown schematically
in Fig. 1(a), and (b). The variation of diode negative
conductance, susceptance, average voltage, and microwave power-generating efficiency with freguency,
dc
current density, and ac voltage amplitude was obtained
with the circuit shown in Fig. 1(a). This method made
it possible to control directly the freguency and ac
voltage amplitude in the calculation and was employed
to economize on computer time. Essentially the same
67
decreases with ac voltage amplitude when the average
current is held constant. Large-signal results at large
currents are more difficult
shown
in
Fig.
1(a)
(voltage
to obtain
with
generator)
the circuit
because
para-
metric [12] efiects occur at large dc current densities.
results were obtained with the circuit shown in Fig. 1(b),
The susceptance shown in Fig. 5 is expressed as a
“capacitance” and plotted in Fig. 7. The value is only
slightly less than the depletion
layer capacitance
(1.9X107* F/cm?), with the largest deviation at the
lower freguencies. The ratio of susceptance to conductance O is shown in
Fig. 8, and the power-generating
if the admittance of the load inductance L and load
efficiencies are plotted in Fig. 9.
conductance G were properly chosen. The coupling
capacitor C served only to isolate the dc and ac portions
of the circuit so that the dc current density could be
directly controlled.
The doping profile for the diode studied in detail is
shown in Fig. 2. The corresponding steady-state elec-
The conductance and susceptance values
as an admirtance plot in Fig. 10. Results for
between 6 and 13.4 GHz and ac volage
between 0 and 38 volts are shown for a
tric field versus distance for this diode is shown in
Fig. 3. To simplify the sign convention on field and
current density, the figures represent a #-v-n-p structure
with the substrate on the left and the diffused junction
on the right. Note that the donor density in the » region
is not negligible, but causes a continuous fall-off of elec-
tric field out to the substrate region. This doping profile
evolved in trial runs to eliminate generation by impact
ionization as the charge pulse of electrons passed
through this region, while still allowing as large an average electric field in the drift region as possible (see
Fig. 12(d) which illustrates this effect).
density of 200 A/cm*.
Also shown
are shown
freguencies
amplitudes
dc current
are constant
effi-
ciency contours. The plot indicates, for example, that
10 percent efficiency is obtained at a freguency of 11.4
GHz with a voltage amplitude of 30 volts, while only
20 volts is required at 8.4 GHz. It was found that the
efficiency increases
with ac voltage
until the
electric
field modulation is so large that, over the negative part
of the voltage cycle, the field dropped so low that the
carrier velocity was less than the saturated value. This
occurs at considerably lower voltage amplitude for the
lower freguencies (less than 7 GHz), and made it very
difficult ro obtain steady-state solutions for these frequencies. Also, note the rapid increase in negative O
(ratio of susceptance to conductance) with ac voltage
ac
amplitude. This would cause the efficiency to turn over
continued until the limit cycle was obtained. The terminal current and voltage waveforms, during a limit
cycle, were analyzed to investigate the variation of
and decrease with further increases in ac voltage amplitude for devices with series lead and contact resistance,
since the diode shunt negative resistance! is reduced by a factor of Q? in converting from a parallel to
a series equivalent circuit.
Jn
a given
run
with
fixed
dc
current
density,
voltage amplitude, and freguency, the program was
diode characteristics at various values of oscillation amplitude. The three fundamental diode characteristics
investigated are negative conductance, susceptance,
and average voltage. Results are shown in Figs. 4, 5,
and 6. The diode's large-signal negative conductance
and susceptance values are effective
values at the volt-
age generator freguency obtained by Fourier analysis of
the terminal current and voltage waveforms. Detailed
results are shown for only one current density because
of the great number of lengthy computer runs reguired
to characterize completely the large-signal behavior.
However, calculations made at other current densities
(100 to 1000 A/cm?£) indicate that the results are gualitatively similar to the detailed results presented for a
current density of 200 A/cm?. The magnitude of negative conductance increases up to about 1000 A/cm?, but
the efficiencies obtained
are less than at 200 A/cm*.
The diode's susceptance is predominately due to the
diode's space-charge capacitance, and therefore decreased with increasing current due to widening of the
space-charge width. The average voltage increases with
current density, but because of the rapid increase of
ionization rates with electric field, the average voltage
Detailed “snap shots” of the electric field, and hole
and electron concentration as functions of distance
during one cycle of steady-state oscillation are shown in
Fig. 11. The “snap shots” arc shown at approximately
one-fourth cycle intervals in Fig. 11(a) through (d).
The figures are selected frames from a computer-made
movie
[6].
A phase plot of the terminal current 7 and
voltage V of the oscillation is included
Points to note are the following.
in the figures.
1) The generation of pulses of holes and electrons
begins where the voltage is a maximum; one-fourth
cycle later, the charge pulses are fully formed and begin
drifting into their respective drift spaces.
2) The holes disappear guickly from the active region while the electrons drift for approximately one-half
cycle and constitute positive particle current while the
ac voltage is negative.
* The caleulation includes all losses that originate between the
electrical contacts, i.e., substrate and avalanche space-charge resistances.
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68
IEEE
TRANSACTIONS
ON
DEVICES,
JANUARY
1969
=
»
o
160,
ELECTRON
o
T
\
»
SUSCEPTANCE
120~
T
ELECTRIC FIELD (VOLTS/Cm)
»1O?
—
>
b
o>
oe
T—T—T—T
140
7.46Hz
6o—
S
Fig. 3.
4
s
6
7
DISTANCE IN MICRONS
0
Static electric field and particle current prol‘le versus distance in dc steady-state Read diode 7
L
20
30
40
AC VOLTAGE AMPLITUDE
50
Diode susceptance (mhos/cm?) as a function of ac voltage
Fig. 5
amplitude for various fixed frequencies. Current density:
A/cm?,
DC VOLTAGE
87
o
CONDUCTANCE
18,
20
30
40
AC VOLTAGE AMPLITUDE
50
60
Fig. 4. Diode negative conductance (mhos/cm?) as a function of
ac voltage amplitude for various fìxed frequencies. Current:
density
200 A/cm*.
200
o
10
l
20
1
30
|
40
AC VOLTAGE AMPLITUDE
L
60
Fig.6, Diode aver e voltage as a function of ac voltage amplitude
for various fixei d freguencies. Current density: 200 A/cm*,
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SCHARFETTER
AND
GUMMEL:
SILICON
READ
DIODE
OSCILLATOR
xJ0-9
1878
2
1.850l—
B4,
24
150
e
w
s xmL
4
r
g
£
s< 1778
%
1.825)—
EFFICIENCY
w
1750~
7.4 GHz,
I
0
25
|
20
AC
Fig. 7.
78+
s0
1725
T
o
100
L
30
VOLTAGE
1
40
L
50
AMPLITUDE
Diode susceptance expressed as an eguivalent capacitance
(depletion layer capacitance
amplitude for various fixed
A/cm?,
L
L
!
20
30
40
AC VOLTAGE AMPLITUDE
60
1.9 X107? F /cm*) versus ac voltage
freguencies. Current density: 200
L
50
60
Fig. 9. Power-generating efficiency versus ac voltage amplitude for
A/cmt.
various fixed frequencies. Current density: 21
1
2
3
SUSCEPTANCE
-60
134
-3ol—
-20l—
30
ovoLTs Ac - 20
o
|
1
20
30
AC VOLTAGE AMPLITUDE
40
Fig. 8. Diode ( (ratio of conductance to susceptance) versus ac
voltage amplitude for various fixed frequencies. Current density:
200 A/em?.
-25
50
Fig. 10.
20
-i5
E
CONDUCTANCE
Diode admittance
54
-5
o
o
o
(susceptance versus conductance) as a
function of frequen cé and ac voltage amplitude, and resultant
efficiency indicated.
urrent density: 200 Â /cm?,
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70
IEEE
127
TRANSACTIONS
50
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DEVICES,
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1969
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Fig. 1. Solutions of hole and electron concentrations, electric field, and terminal current and voltage (values indicated by &
on phase plot) at various polats in time for the diodeoperating at a frequency of 12.4 GHz, current density of 200 A/erm’,
an efficiency of 12 percent.
3) For the next one-fourth cycle, the remnants of the
electron charge pulse are swept out of the picture as the
voltage again approaches its maximum value.
4) The displacement current is guite large and has
an appreciable swing into the conventional forward
direction, while the terminal voltage always remains in
the conventional reverse polarity.
5)
The behavior is as predicted by Read, except that
for silicon diodes an extended avalanche region is reguired relative to the drift region to obtain the magnitude of charge pulse necessary for sufficient modulation of diode voltage and particle current for efficient
oscillations.
Efficient oscillations were also obtained at higher dc
current densities by trial and error choices for the load
conductance G and load inductance L. “Snap shots” of
the oscillation at a dc current density of 1000 A/cm? are
shown in Fig. 12(a) through (d). The largest efficiency
obtained for this bias (9 percent) occurred at an operat-
ing freguency oí about 13.4 GHz. Note that the carrier
concentration scales differ by a factor of 5 between Figs.
11 and 12, as do the dc current densities. However, the
large-charge pulse, for the 1000 A/cm? case, moving
through the drift space results in a significant value of
peak field near the substrate just as the electrons approach this region. If this peak field is too large, genera-
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SCHARFETTER
60
AND
GUMMEL:
SILICON
READ
DIODE
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OSCILLATOR
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9 20
v
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z
50
s0
Ha0,
2
2
<40
U
*E |
e 300
g
71
HOLES ~_]
PLOT
"
40,
2
240
M
50|-
e
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d
s
i
q20 £
g
3
& 30f
$
F
z
&
3920
jio
ELECTRIC FIELD
2
d
s
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s
£
2E
9
-
s
H
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ol
-fD
2
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dao _
v
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30-
S
w
H
20 o
201
d
Hi0
E”
“S4
@
1o
ND”
o
12
Fig. 12.
3
4
5
6
7
DISTANCE IN MICRONS
(b)
8
AN0
9
o!
o
-
1
-
2
a5
6
7
DISTANCE IN MICRONS
(d)
9
0
o
Solutions of hole and electron concentrations, electric field, and terminal current and voltage (values indicated by
& on phase plot) at various points in time for the diode operating at a freguency of 13.4 GHz, current density of 1000
A/cm*, and efficiency of 9 percent.
tion by impact ionization occurs, and the subseguent
drift of a hole charge pulse toward the p region would
constitute positive particle current while the voltage
was also positive, a very unfavorable phase relation.
This effect was worse in an earlier doping profile (not
shown), and was eliminated by altering the profile to
provide a continuous fall-off of electric field out to the
substrate
ticle current is not a sguare wave during the half cycle
(see Fig. 3).
The nearly ideal (classical Read
obtained by subtracting the displacement current from
the total current. At the largest oscillation amplitude,
the diode “capacitance” changes parametrically [12]
and makes an accurate separation of the current components difficult, The result shown in Fig. 13 is for a
voltage amplitude of about 17 volts. However, the
waveforms for larger amplitudes are similar. The par-
[1]) phase relations
obtained between diode voltage and particle current is
illustrated in Fig. 13. In this figure, we plot the waveforms in time of the particle current and terminal voltage for an ac steady-state solution at 11.4 GHz and dc
current density of 200 A/cm?. The particle current was
over which the voltage is negative, but has a hump resulting from the extra component of particle current
which flows until the holes that were generated along
with the electrons are swept into the p layer. For large
amplitudes, the hump becomes more pronounced and
the particle current bottoms [all carriers are swept out
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72
IEEE
TRANSACTIONS
ON
ELECTRON
DEVICES,
JANUARY
1969
APPENDIX
DIODE
VOLTAGE
SEMICONDUCTOR DEVICE ANALYSIS
CoMPUTER PROGRAM
INTRODUCTION
This Appendix
consists of three sections. Section
I
describes the device physics included in the large-signal
calculation and outlines the solution procedures with-
out detail. Section
II describes in detail the time ad-
vancement technigues and the mathematical formulation employed in the computer calculation. Section 1II
is a detailed discussion of the matrix inversion subrou-
PARTICLE
CURRENT
LIMIT CYCLE
FREQ * 11.4 GHZ
tine developed
specifically for the time evolution com-
puter program.
I. MATHEMATICAL
A.
TIME
Fig. 13. Diode voltage and particle current versus time for one
cycle of steady-state oscillation.
Charge
MODEL
Transport Eguations
The distribution and motion of carriers within a onedimensional semiconductor device structure can be obtained by solving three basic eguations: 1) the continuity eguation for holes, 2), the continuity eguation
for electrons, and 3) Poisson's eguation:
as shown in Fig. 11(a)] over an appreciable part of the
cycle.
(13)
The results of the calculation presented in this section
have shown that a properly designed silicon Read diode
oscillator is operable, with efficiencies from 9 to 18
percent, over an order of magnitude variation in dc
current. The freguency of operation, amplitudes of
voltage and current oscillations, order of magnitude of
(14)
and
ôE
g
.
—=—(p—n+Np—Ni)
dc bias, and efficiency are in fair agreement with the
prediction of the relatively simple design theory pre-
sented in Section II.
dx
where
ôp
J, — quppE — kTu,—
IV. CONCLUSIONS
ôv
Operating characteristics of a silicon Read diode were
obtained from large-signal computer calculations of the
evolution in time of the diode-resonant-circuit system.
The values of ionization rates [3] and carrier velocities
[8], [9] appropriate for silicon were used in the calculation.
Self-consistent
solutions
were
obtained
for
the
eguations describing carrier transport, carrier generation, and space-charge balance within the diode, and
which simultaneously satisfied the boundary condition
imposed at the diode metallic contacts by the resonant
circuit.
Detailed
electron
solutions
€
.
(15)
were
concentrations,
presented
electric
of
field,
the
and
hole
and
terminal
current and voltage at various points in time during ac
steady-state cycles of oscillation. The largest efficiency
obtained (18 percent) is in good agreement with the
approximate calculation, which includes the appreciable width of the avalanche region reguired for silicon
devices. Efficient operation (9 to 18 percent) was obtained over a range of dc current density (100 to 1000
A/cm?) and freguency (6.4 to 13.4 GHz).
ôn
Ju = gunnE-r kTu — ôx
(16)
an)
Boundary conditions are imposed at the contacts by introducing the appropriate restrictions in (13) through
(17). For example, current boundary conditions for a
p-n device are introduced by reguiring that (17) is
egual to the terminal current density at the » contact
and that (16) is egual to the terminal current density at
the p contact. Voltage boundary conditions are imposed by reguiring that the integral of E(x, f) over the
interval between the two contacts eguals the total
voltage. In addition, the electric field at the two end
metallic contacts is assumed to be zero. Initial values
for the hole and electron densities are either furnished
by
a previous
run or are given
by
the guiescent
zero
bias solution, i.e., the solution when
(18)
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SCHARFETTER
AND
GUMMEL:
SILICON
READ
DIODE
OSCILLATOR
In general, the functions g, u,, 4, vary at each point
in the device according to the value of m, p, and E at
that point.
B.
tinuity eguations for each increment:
d,
Eg(.\') =g(V)
Carrier Generation-Recombination Expressions
The carrier generation term g is composed of two
components: 1) carrier generation and recombination
through defects, and 2) impact or avalanche ionization.
Hole electron generation and recombination through
defects are represented
by
a Shockley-Read-Hall
(SRH) single-level model which characterizes defects
with neutral and single-charge states. The generationrecombination rate through a single-level center is
given by
pn—no
8=
The
upon
tron
given
(19)
Tpo(n 4 m) + Tso(p + P9)
impact ionization rates are strongly dependent
the electric field intensity and the hole and eleccurrent densities. These generation terms are
by
1
o= (an(E)|1| + aK(E)|J,)
(20)
73
= (J,(M) —J, (M — 1))/Ax
dn
i
” (V) > g(Y) + (Ju(M) —J(M—1))/Av.
= 2.25 X 107 exp (—3.2 X 105/F)
Ax
where the Mth mesh point is located midway between
the major mesh points N4+1 and N. It is customary to
employ next the standard difference approximations in
the current density expressions (16) and (17) and substitute these results in (25) and (26). However, it can
be shown
stability
points
that this procedure
whenever
exceeds
the
2k7/g.
Rather,
Holes
Electrons
N
S
(E/a)?
A
480
4X10%
816.1X10°
1400
3X10'¢
3503.5X10*
¥ F)
(23)
F
B
1.6
2.5X10*
8.8
and
inmesh
(17)
are
J.0) = zan) — n
”
77
DM
(1.0 — exp (— E(M)Ax)
.
+
——
(29)
—,
(1.0 — exp (E(M)As))
These eguations provide numerically stable estimates
of the current density under all conditions. If the intermesh point voltage is small, these eguations approach
the standard difference relations; whereas when the
voltage change is large, they approach the drift current
density at either mesh point N or N+1.
7.4X10%
(24)
D. Solution Procedures
Because of the nonlinearities in the eguations describing the hole, electron, and field distributions, obtaining a transient or even a steady-state dc solution
poses a very difficult numerical problem. The structure
to be analyzed is first subdivided into a number of small
cells. The eguations are then normalized to reduce
redundant coefficient calculations, and standard difference approximations are used
tial derivatives in Poisson's
(16)
between
PN + 1)u (M)
]
(1.0 — exp (E(M)ax))
(22)
+ (E/B)*
o
change
T4() = EO I:(x—.o — exp (—E(M)A))
(21)
following expression:
(/a =1 *(m)*(ïq
leads to numerical
voltage
treated as differential eguations in p and m with J,, J,,
H M», and E assumed constant between mesh points.
The solution of these differential eguations then relates
J» and J, to the other variables:
It is necessary to include in the analysis the variations
of mobility (u) with the electric field (E) and the
ionized impurities density (Np).
The theoretical mobility is approximated by the
Np
2
= L () — n(8) 4 No(¥) — Na(8))
Mobility Expressions
;
;
P(N)m;(M)
an(E) = 3.80 X 10* exp (—1.75 X 10¢/E).
C.
(26)
E(M) — E(M — 1)
where a, (E) and a,(E) are given by the relations
an(E)
(23)
to approximate the spaeguation and the con-
E.
Definition of Symbols
f, n
E
g
Jp J»
hole and electron concentrations, cm^*
electric field intensity, V /cm
electronic charge, 1.602 X 10~ coulombs
hole and electron current densities,
A/cm?
g hole and electron generation-recom-
bination rates, carriers/cm*-s
gr impact
ionization
generation
rate
for
holes and electrons, carriers/cm*:s
g: hole and electron generation-recombination rates through a single-level
trapping center, carriers/cm*-s
€ dielectric constant
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74
1EEE
k Boltzmann
constant,
1.38044X10-%
JK
TRANSACTIONS
by
the
ON
closed-form
concentration of donor
impurity atoms, cm^*
ay(E), as(E)
P
m
and
acceptor
hole and electron mobilities, cm?/V-s
low-field low-doping hole mobility in
silicon, (480 cm?/V -s)
low-field low-doping electron mobility
in silicon, (1400 cm?/V-s)
ionization coefficients for holes and
Eguation
increment
(33).
JANUARY
Written
in
1969
more
¢ in
II. VECTOR-MATRIX
level of the
FORMULATION
time.
When
we
have
solved
it,
we
increment the vector y by the solution ôy. To get a
better intuitive understanding of what we are doing, let
us move the term containing J/t in (36) on the righthand side:
ôy(t) = (f r %U)I.
hole and electron concentrations in the
coincides with the energy
single-level trap.
(36)
— f*t.
(36) is the vector eguation we solve for each
electrons, carriers/carrier-cm
conduction band when the Fermi level
A.
expression
(l —%)ôy
cm™?
Ma
DEVICES,
conventional form,
1" absolute temperature, *K
Np total ionized impurity concentration,
M
ELECTRON
@37
At time 0, the slope ŷ is f(ys). At time ¢, it is approximately fo+Mdy.
The
ratio
of the
finite
differences
ôy(f)/t corresponds to the slope of the straight line
through the points 0 and f. What we are doing is
propagating the system according to the time derivative
at some advanced time f,—t/2, rather than at the ini-
Time Advancement
We represent the hole and electron concentrations by
the two-component vector y. The system of eguations
then reads
ŷ > f),
(30)
i.e., the time derivative of y eguals some nonlinear
functionsf of y. We denote y at the beginning of the
time interval under study by yo, and denote the deviations of this value as time increases by ôy(f):
y(0) = yo+ ôy(D.
G
tial time. The term in ôy on the right-hand side of (37)
represents a feedback that gives the system stability.
It is worth noting that the expansion of f with respect
to y (the derivative matrix M) is correct to first order,
but with regard to time it is correct to second order in ¢.
The method of propagation (36) time evolves the system accurately for small time steps f. However, if we
are near an eguilibrium state and are interested not in
the time evolution, but only in the equilibrium solution, time steps ¢ of infinity can be taken with full
feedback, i.e.,
(1 — Mi)by = f-t.
For small deviations ôy, we expand f(y) as
83 — f(yo) + Môy.
(32)
M is the matrix d//dy and represents an integro-differential operator. Now, if M, f, and 8y were ordinary
B.
The Derivative Malrix
We define the vector y as having components of hole
concentration, electron concentration, and electric field:
numbers, the solution to (32) would be
ôy(t) = (M — 1)f/M
(33)
Jx —
or
»)
(AHOL(N)
|:
— | AELE(N)
lys
ôy(t) = f1(1 + Mt/2 + (MD*/6 + -+
-).
(34)
Note that if the physical situation is one in which we
are near a stable equilibrium state, then the eigenvalues
of the matrix M must be negative so that as time goes to
infinity, ôy reaches a finite value.
For small but finite time steps, we introduce the inverse of the first two terms of (34) to obtain our solution
to (30) as
st
-
B0 = =
(85)
2
'The operator M in the denominator roughly corresponds to integration and as f goes to infinity, we reach
an asymptotic value within a factor of 2 of that given
and the right-hand
the vector f:
ï
side of the continuity equations as
=)~ Gt )
LN
The vector equation
(38)
|AFIELD(M))
we solve for each
(39
time increment
f (36) is
2
WN
My | G)x > 2(fo)s + (Af)
(40)
where Af, represents changes in fy at the terminals and
where we define the derivative matrix
Iy
My ===
N
ay
(41)
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SCHARFETTER
AND
GUMMEL:
SILICON
READ
DIODE
OSCILLATOR
and
The matrix elements of M are defined by the expanded
form of (40) shown below:
XLG,N)*ôH(N
et (e* DO (a,
*
B(,
;
y1
”
Rl
+
(42)
|
FLO,
matrices
XL, XM, and XR are obtained by taking the partial
derivatives of fy with respect to HOL and ELE at mesh
points N—1, N, and N+1, respectively, while the two
component vectors FL and FR are obtained by taking
the partial derivatives of fy with respect to FIELD at
M —1, respectively.
N) = 6F(M
$F(M)
— 8F(M
DY(X)
Input
right-hand side RH(1) of the first continuity eguation,
and the particle currents at the right contact enter into
the right-hand side RH(LR) of the last continuity
eguation. Since the total current at the contacts is
known, continuity eguations 1 and LR contain three
boundary conditions. The fourth boundary condition is
current
will
be
described
in
Section
IlI-C
of
their
reguired
M =0,LR
[XLly
N-2,LR
[xM]y
N =1,LR
[XRly
N=1,ILM
(RH)y
(FL)y
(FR)y
N =1,LR
N-2LR
N =1,LM.
B.
(LM =LR-1)
Details of Inversion
We represent the output quantities ôH and ôE by the
vector
A. Input-Output
y
as
and similarly input guantities RH,
,
FL, FR as
RHO, N)
XL(1, X) * 8H(N — 1) + XL(2, N) * ôE(N — 1)
+
XM(1,N) * ôH(N) + XM (2, N) » 6E(N)
+
(43)
XR(1,X) * 8H(N + 1) + XR(2, N) * 8E(N + 1)
(RH)x = (RH(Z, .\'))
FLO, M)
Wi
= (n(z, M))
s
_ (FRO, M)
F
k
= RH, N),
as
(note ôF(0) and ôF(LR) are zero).
SH(N)
FL(1, N) * 8F(M — 1) + FR(1, N) » 6F(M)
are
N =1,LR
N =1,LR
M -A,LM
G)x = (ôE(.Y)
solves the
range
(45)
their range are as follows:
Appendix.
We describe a computer subroutine which
following three eguations:
— 1))/2.
DXy
ôHx = Y(1, N)
ôÊx = YO, N)
ôFw — YG, M)
a
this
+ DX(M
and
Output quantities and
field of zero at one contact forces a field of zero at the
other contact. The method for solving the discretized
continuity eguations will be described for the case of
terminal-current boundary conditions. The modifications to this method when boundary conditions are in
terms of terminal voltage or a mixture of voltage and
= (DX(M)
guantities
that the electric field at the contact is zero. Since overall
the eguations,
— 8E(N)) » DY(N)
follows:
the particle currents at the left contact enter into the
into
— 1) = (H(N)
where
for the guantities §H (N), 8E(N), and ôF(M). The guan-
is built
1) + FR(2, N) * 8F(M)
where N ranges from 1 to LR, and
tities ôH and ôE are the changes in the hole and electron
concentrations, while ôF is a normalized electric field;
ôF is in units of carrier concentration. The boundary
conditions have been included in the eguations, i.e.,
neutrality
—
= RH(2, )
INVERSION SUBROUTINE
This section describes in detail the method for solving
the discretized continuity and space-charge eguations
space-charge
(
+
subroutine described in Section III. The 2X2
MATRIX
A
XRG,N)*ôH(N + 1) + XR(4, N) *ôE(Y 4 1)
Vector eguation (42) is solved by the computer program
III.
* ôE(N — 1)
XM, N) * 8H(N) + XM(4, N) * 6E(N)
(RH)x — 2(fo)x.
mesh points M and
— 1) + XL(4, N)
« (¥
+ (FL)x * ynt—y + (FR)m * yu = (RH)x
where
75
The
= (FR(Z, M))’
input guantities XL,
X M, XR
are represented by
matrices [XL], [XM], [XR]:
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76
IEEE
—XL(1,N)
XLG,N)
vl
XLGa,N)
XLG,N
TRANSACTIONS
Note
this notation,
(43) through
(45) can be written
|XZ]w * ()xor r [XM]y * (y + [XR]x * Gy
Eguation
(46):
XLy
(45)
— 1) + (FR)y * 8F(M)
can
be
used
as)
= (RH)y.
to eliminate
DEVICES,
JANUARY
ôF(M—1)
in
1969
that (48) can be written as
"
— (UF), * ôF(L)
as
+ (FL)y * 6F(M
ELECTRON
2
l
(ME),
= (W'«
1&1 — EXRl. G
and similarly for [X M]y and [XR]y.
With
ON
and
(45) as
ôF(L — 1) — ôF(L) — Í8H(L) — ôE(L)) » DY(L); / (53)
therefore, once we have [UM], (UF), and (Z) at mesh
points 1 to LR, we can work backwards using (52) and
53) to find
(v + {[XM]y — [UFL]s] * 0y
+ [XR]y * (y)ssa F [(ER)y + (FL)y) * 8F(M) = (RH)y
where
(EE)L
and 8Fz—y for L=LR
[UFL]y = [
FL(1, N) — FL(1, N)
FLO, N) — FLO, .\’)] *DYO
GD
We assume that we can remove the (y)y_; term in (47)
and write the resulting eguation at mesh point N—1 as
[UM]x-— * (y)s— + [U R]w—s * G)x
as)
+ (UF)x— * 6F(M — 1) — (Z)x—.
Recursion relationships for [UM],
[UR], (UF), and
(Z) are obtained as follows.
Eguations (47) and (48) are combined
to eliminate
(9)x— terms as
{(XM]y — [UFL]s — [Tly * ÍUR]w—i) * (v
+ [XR]s * )1 F H FR)u F (FL)w) * bF(m)
— [T]s * (UF)x— * 5F(M — 1)
(a9)
— (RH)x — Ty * (Z)y—i
[7]y = [XL]y * [UM]yE.
The term in ôF(m—1) can be eliminated in (49) by
(45), and therefore by comparing (49) with (48) evaluated at mesh point N, we get:
[UR]s = [XR]x
|UM]x — [XM]x — lUFL]x — [T)y * [UR]xa
DY(N) — DY(N)
I:Dym — DY(V) ] * [T]s * (UF)x
(Z)x = (RH)x — [Ty * (Zx—)
(UF)y = (FR)x + (FL)y — [Ty * (UF)x—x.
(50)
Starting values for [UM], (UF), and (Z) are obtained
by comparing (48) (at N=1) with (46) (at N=1) with
the term in ôF(1) eliminated by (45):
FR(1,1)
[UM), = (XM, + [ FR(2,1)
=)
(Z): > (RH),
— FR(1,
1)
— FR(2, 1)
:I. DY (1)
(51)
- - - 1.
'The computation procedure is as follows.
1) Compute starting values from (51).
2) Compute values at the remaining mesh points
(2, LR) using (50).
3) Use (52) and (53), starting at mesh points LR
(note (y);»31 and ôF(LR) are both zero) and working
through mesh point 1 to obtain the values for ôH, ôE,
and ôF.
C. General Boundary
Conditions
In the discussion thus far, it has been assumed that
the desired boundary conditions could be incorporated
into the input coefficients. The terminal currents entered into the right-hand side (RH)y at the contacts
(N=1) and (N=LR). More general boundary conditions can be incorporated by using an economical
superposition process.
With the time advancement process described in
Section II, we want to solve the eguations
[z-»n])«e-
(51)
where f is the step in time and M is the derivative
matrix. If the terminal currents are changing with time,
then f is fo, the right-hand side of the continuity eguation at ¢=0, plus one-half of the change in terminal
current (the extra term will appear in only two egua-
tions, mesh points 1 and LR). Let us multiply (54) by
2 and write it as
2
[F+a]mr=2en+w,
)
where Af represents the change in terminal currents in
the time f. Equation (55) is identical to (46) and could
be solved by the method discussed previously in Section
I11-A of this Appendix if the change in terminal currents
(Af terms) are known. What is always
terminals is one of three possibilities:
known
at the
1) the terminal current as a function of time is an
independent variable;
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SCHARFETTER
AND
GUMMEL:
2) the terminal voltage
independent variable; or
3) an eguation relating
voltage is known.
The general solution to
obtained numerically using
SILICON
READ
DIODE
as a function of time
is an
77
voltage per change
in terminal current Av.. If the inte-
gral of the electric field component
terminal current
to terminal
(55) for Af arbitrary can be
superposition. Let us assume
that our final solution will be the sum of two solutions:
() > () + (o).
(36)
These solutions have the following interpretation:
(ye)
represents the solution for Af=0, that is, for no
change in terminal current over the time step f;
(y.) represents the additional solution for a change
in terminal current.
'To see this more clearly, let us write (55) as
|M]y() = (RHi)s + (RH.)y
where the quantity
OSCILLATOR
(57)
2/1--M in (55) has been written as
[M]y in (57) and 2fo as (RHs)x and Af as (RH.)x.
Eguation (57) has (RH.)y zero everywhere, except at
mesh points 1 and LR. Therefore, we solve (57) using
the method described in Section III-A of this Appendix
with (RH)y — (RH,)y. This yields solutions (yo)x. Next,
we solve (57) with (RH)y > (RH.)y; this yields solution
(y.)w. Since we do not know the change in terminal
current AJ, we obtain solutions (y.)y for a unit change
in terminal current. We then integrate the electric field
component of (y.)y to obtain the change in terminal
of (ys)x is called
AT, then the change in terminal voltage ATV is related
to the change in terminal current AJ by
AV = AV5 F Av. * AJ.
(58)
REFERENCES
(1) W. T. Read, “A proposed high-freguency negative-resistance
diode,” Bell Sys. T&ch. J. vol. 37, pp, 401-466, March 1958.
[2) H. K. Gummel and
Scharfetter, “Avalanche region of
IMPATT diodes,” BellSy < Tech. J., vol. 45, pp. 1797-1827,
nd W,
Wiegmann, “Ionization rates of holes and electrons in silicon,”
Phys. Rev., vol. 137, pp. A761-A773, May 1964.
(4] M. Gilden and M. B, Hines, “Electronic tuning effects in the
read microwave avalanche diode,” JEEE Trans. Electron Devices, yol. ED-13, pp. 169-175, January 1966.
[S] T. Misawa, “Negative resistance in — junctions under avalanche breakdown conditions, parts | and lI,” IEEE Trans.
Electron Devices, vol. ED-13, Pp. 137-151, January 1966.
(6] D. L. Scharfetter and H. K. Gummel, “Design of Read diode
oscillators,” presented at the 1966 IÊEE Solid-State Device
Research Conf., paper 111 b-4,
17] J. C. Irvin, “Resistivity of bulk silicon and of diffused layers in
silicon,” Bell Sys. Tech. J., vol. 31, pp. 387-410, March 1962.
8] A. C. Prior, “Field dependence of carrier mobility
in silicon and
ïrmnnium," J. Phys. Chem. Solids, vol. 1
pp. 175-180,
anuary, 1960.
191 . E. Seidel and D. L. Scharfetter, “Dependence of hole velocity
upon electric field and hole density for P-type silicon,” J. Phys.
Chem. Solids, vol. 28, pp. 2563-2573, 1967
{10] W. Shockley
and W. T. Read, Jr., “Statistics
of the recombination of hole and electron,” Phys. Rev., vol. 87, pp. 835-842,
September 1952.
(11] R. N. Hall, “Electron-hole recombination in germanium,” Phys.
Recs vol. 87, p. 387, July 1952,
[12] B. C. DeLoach, Jr., and R. L. Johnston, “Avalanche transittime microwave oscillators and amplifiers,” JEEE Trans.
Electron Devices, vol. ED-13, pp. 181-186, January 1966.
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