I E E E TR-ANS.ACTIONS ON ELECTROS DEVICES, VOL. ~ ~ 1 - 1 so.
6 , 1, J.ASLARY 1969
Large-Signal Analysis of a Silicon
Read Diode Oscillator
Abstract-Thispaperpresentstheoreticalcalculations
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
startingdesign.Thisdesignisthenmodifiedtoachievehigher
efficiencyoperation as specificdevicelimitations
arereachedin
large-signal(computer)operation.Self-consistentnumericalsolutions are obtained for equations describing carrier transport, carrier
generation,andspace-chargebalance.Thesolutionsdescribethe
evolution in time of the diode and its associated resonant circuit.
Detailed solutions are presented of the hole and electron concentrations,electric field, andterminalcurrentandvoltageatvarious
points in time during a cycleof oscillation. Large-signal values of the
diode’snegativeconductance,susceptance,averagevoltage,and
power-generating efficiency are presented as a function of oscillation
amplitudefora
fixed averagecurrentdensity.Forthestructure
studied,thelargest
microwavepower-generatingefficiency
(18
percent at 9.6 GHz) has been obtained at a current density of 200
A/cmZ, but efficiencies near10 percent were obtained over a rangeof
current density from 100 to 1000 A/cm2.
s
I. INTRODUCTIOS
The aim of thedetailedcalculation,inadditionto
the enhancement of our understandingof the oscillation
mechanisms, was to find conditions under which high
efficiency and power outputcan be obtained,andto
A
establish
theoretical
limits
for these
quantities.
rigorous optimization, however, was not attempted in
thisexploratory n-ork. This would have had to start
with carefully formulated constraints and would have
constituted a very large-scaleproject.However,the
largest efficiency obtainedinthedetailednumerical
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 . VoltageDriven
To study the performance at a given frequency, 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 frequency
is obtained as a function of driving voltage amplitude.
Likewise, the ac power delivered by the diode into the
voltage generator is obtained.
E L F - C O N S I S T E N T large-signalnumericalsolutionsareobtained
for equations whichdescribe
carrier generation and space-charge balance in
a
silicon p-n-v-n Readdiode
[ l ] microwaveoscillator.
The solutions describe the evolution in time
of the diode
and associated circuit (see Fig. 1). The doping profile
of the diode (see Fig. 2) was evolved starting Ivith design considerations discussed in [2], and reviewed here, B . Free-RunningOscillator
and modified afterinitialresults. for improvement of
For another series of calculations,thediode is imefficiency. From the numerical solutions, the large-sigbeddedinaresonantcircuit
[see Fig. l(b)] and pernal operating characteristics of the diode and the
effi- forms as a free-running oscillator. In as much as the ac
ciency atthefundamentalfrequency
(defined as ac voltage in the oscillator case is nearly sinusoidal, both
powerdelivered
bythediodedividedbydc
power calculationsshould,andindeeddo,givecomparable
dissipated) are computed.
results.
Read in his original paper
[l ] suggested that an effiIn Section 111,n.e present the large-signal operating
ciency of 30 percent should be obtainable in
a silicon characteristicsanddetailedsolutions.Theoperating
diode.Thiscalculation,however,neglectedthefinite
characteristics,
i.e.,
admittance
parameters,
average
width of the avalanche region, and neglected the differvoltage, and power generating efficiency, are shown as
ence in hole and electron ionization coefficients in
sili- a function of diodeacvoltageamplitude
in Figs.4
con. A modification of Read’scalculation, which in- through 9. Fig. 10 is a diode admittance plot (small and
cludes the finite width of the avalanche region and the
large signal) which sho\vs the variation of conductance
experimental values of ionization coefficients for silicon andsusceptancewithoscillationamplitudeandfre[ 3 ] , is presentedinSection
11. I t isfound thatthe
quency. Also indicatedarethe
power generating effiavalancheregionmusthaveanappreciablewidthfor
ciencies. Fig. 10 is a compact presentation of the importhe diode tohave
high
power
capabilityandhigh
tant results of thispaper.Detailedsolutionsarepreefficiency. A theoretical efficiency of theorder of 15 sented of the hole and electron concentrations, electric
percent is obtained for conditionswhichshould
ap- field, and terminal current and voltage at various points
proach maximum power output.
in time during a cycle of oscillation. These solutions are
sholvnin
Fig. 11 ( J d , =200.4/cm2,frequency = 11.4
Manuscript received June 3, 1968.
GHz)
and
Fig. 12 ( I d c = 1000 -4,’cm2, frequency
The authors are with Bell Telephone Laboratories, Inc., hlurray
Hill, N. J .
= 13.4GHz).The
d3,namics ofcharge-pulsebuildup
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
SCHARFETTER . W D GUMXIEL: SILICOK RE;\D DIODE 0SCILL.ATOR
65
one-half the average voltage l,,D developed across the
drift region [ l ] , [ 2 ] . L\tthedriftfrequency
(f~=onehalf theinversetransittime
of thedriftregion),the
motion of Qmaxresults under favorable conditions in an
ac particle current \vhich is = 180 degrees out of phase
\vith the ac voltage across the diode. The average of the
particle current is the dc current Jdc. The particle current slving is, therefore, a t most from zero to twice the
dc current. For a square wave of particle current and a
sinusoidal variation of drift voltage, both with magnitudeandphaseasdescribedabove,themicro\vave
poi\-er-generating efficiency is
+TrTIG
(a)
(b)
\\-here the total dc voltage is divided betxveen the part
developedacrosstheavalanche
region I'dcA andthe
part developed across the drift region
T-~,D. S o t e t h a t
\ve are neglecting any appreciallle average ac
po\ver contribution from the avalanche region; this is because the
IO"
avalanche region voltage is inductively reactive relative
to the particle current [4]. The displacement current is
capacitively reactive relative to the diode voltage and
therefore contributes no average ac pon-er. T h e use of a
simple equivalent circuit [4] for the avalanche region,
\vhich is derived from small-signal considerations, may
of course have no validity under large-signal conditions.
However, n-e will use this approach as a starting point
for a simple 'design theor>-, and compare detailed
numerical calculations \\-ith the simple treatment.
I n Read'scalculation, it \\-as assumedthat no dc
po\ver loss \\-as associated \\it11 theavalancheregion
N+
N
P
since it n-as considered small in extent cornpared to the
drift
region, and that such an avalanche
regioncould
1
0
~
~
I~
I'
11
1
I
1
1
1
2
3
4
5
6
7
8
9
1
0
produce an ac particle current amplitude equal to the
DISTANCE IN MICRONS
dc current \\-henoperated at the drift frequency and
Fig. 2 . Netimpurityconcentration
of Readdiode
7A versus distance.
operated I v i t h ac voltage amplitude equal to one-half
thedcvoltage[sameassumptionsas
used above in
deriving ( I ) ] . Since TTdcA a-as neglectedcompared
to
i n theavalanche region anddriftthroughthedrift
1 7 d c D , Read
predictedan
efficiency of 1,'n or abut
region is in qualitative agreement \\it11 Read's original 30 percent.
prediction of operation.
Small-signalcalculations [ I ] , [ 2 ] , [4], [SI shou. t h a t
Section 11: presentsconclusions,andtheAppendix
aresonancefrequerqf x is associated \\it11 an?' avdescribes the time evolution computer program.
alanche region, and that it increases as the square root
of dccurrent. To obtain small-signalnegative
resis11. SIMPLIFIED DIODE DESIGN
tancerequiresoperationahovethisfrequency.
Ho\vIn this section, we extend Read's efficiency calcula- ever, operation of an avalanche region a t a frequency
tion toinclude
thedcvoltagedevelopedacrossthe
\vel1 above its resonance frequenq. \vas found to result
avalanche region, and to allow for the unequal ionizainvery inefficient oscillationsinalimitednumber
of
tion rates of holes andelectrons, as appropriate for exploratorylarge-signalcalculations
t h a t \\-erecomsilicon [3]. As pointed out by Read, high-power operapleted on a variety of Read diode structures. Small-sigtion requires the generation of as large a charge pulse
nal calculations in the frequency domain [ 2 ] showed a n
Qmaxas possible in the avalanche region, without a reoptinlum growth factor for small-signal oscillations ocduction of the electric field in the drift region below t h a t curringwhentheoperatingfrequency
\vas about 20
required for velocity saturation when this charge moves percent higher than the resonance frequency. Therefore,
throughthedriftregion.
The motion of Qmax through \\-e take as a designcriterionthattheavalanchefrethe drift region results in an ac voltage amplitude about quency of a Read diode should be related to the drift
Fig. 1.
Diode and associatedcircuit.(a)Voltagedriven.
running oscillator.
(b) Free-
c
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
JANUARY
DEVICES,
?’R;\NSXCTIONStON
ELECTRON
IEEE
66
frequency of the diode by the following condition:
f D
s 1.2fR.
(2)
1969
(2TfR)’
The small-signal constraint above, coupled with largesignal conditions imposed on the drift space
to be discussed below, results in a tentative design for the Read
diode n-hich is analyzed numerically in this paper.
Thesquare of theavalanche regionresonancefrequency is proportional to the dc current density J d c r the
carrierscatteringlimitedvelocity
z J , andthepartial
derivative of theaverageionizationrate
B.4 forelectrons in the avalancheregion with respect to the average
electric field EA.-4n approximation for the relationship
is [ I ] , [21, [41
( 2 q f ~=
) ~2B-4‘7’,lJdc/’~,
(3)
\\-here E is the dielectric constant, and
BA’ the partial
derivation of B A \\-it11 respectto E,. Thedcvoltage
developed across the avalanche region is defined as
I’dr..I
=
EALA,
(4
and the dc voltage developed across the drift region
similarly defined as
L’dcD
= EDLD,
\\here L A and L O arethe effectiveIvidths
alancheanddriftregions.l17iththedefinition
frequency
is
( 3
of theavof drift
by ( l o ) , (8), and
evaluated for theconditionsgiven
( 7 ) , is
2
x6x3x
VdcDfD2/lidcA,
(1
1)
and with the condition onfR andfD given by ( 2 ) results
in a relation between VdeA and I”dcD:
VdcA 1
: 1.1VdcD.
(12)
This result is only qualitatively correct, sinceit is based
on a combination of small-signalresultsfortheavalanche region, and large-signal constraints on the drift
region. Furthermore, the factor 2 in (3) a n d ( l l ) , t h e
factor 6 in (10) and ( l l ) , and the factor 3 in (9) a n d
(11) are all approximate, and depend to some extent
upondcbias,frequency,andoscillationamplitude.
HoLvever, the qualitative result is important: the avalancheregionvoltage,for
a Readdiodedesigned
for
efficient high-poxver 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).
Xote that for Ge, GaAs, or other materials with nearly
equal ionization rates, the factor
3 in (9) becomes a 1
and 1’dcAs+Vdc~, hence the theoretical efficiency is of
the order of 23 percent.
111. LAKGE-SIGSAL
CALKLATIOX
Inthissection,
we discussthepreciselarge-signal
operatingcharacteristicsof
a particularReaddiode
oscillator, as obtained by a numerical calculation perfD = ?‘,1/2LD,
(6) formed on a high-speed digital computer. Details of the
computer
program
are
discussed
in the
Appendix.
1 7 d c D can be expressed as
BrieflJ-, theapproach is toobtainself-consistentnuVdcD = ED?‘al//’2fD.
( 7 ) mericalsolutionsfortheequationsdescribingcarrier
transport, carrier generation, and space-charge balance
For 50-percent voltage modulation ( T T a c ‘ v $ 1 7 d c D ) , Qnlas
in one-dimensional
a
semiconductor
structure.
The
from Gauss’ theorem is about ‘ v d ? D / 2 . The ac particle solutions describe the evolutionin time of the diode and
current (\vhich
for
100-percent
current
modulation
itsassociatedresonantcircuit.Basically,theprogram
equals the dc current) equals
Q m a x f D ; therefore, the dc
solves the follolving problem a t variousinstances of
bias current densit)- for efficient operation of the drift
time during a cycle of oscillation. Given the instantaspace is proportional to frequency:
neous distribution of the
hole and electron concentrations
and
terminal
boundary
conditions,
how
will the
Jdc = € E D ~ D / ~ .
(8)
carriers move in time, i.e.,
how will the system, diode,
For ionization rates appropriate for silicon [3], we find, and circuit time evolve? Given the carriers, the program
for junctions \\-it11 breakdoum fieldsbetlveen
3 and determines the electric field by Poisson’s equation, and
4 X lo5 \‘:’cm, that the integralof the electron ionization knolving the electric field and carrier concentrations, it
rate n-it11 distance is not unity, as is the case for equal
obtainstheinstantaneous
hole andelectronparticle
rates, but about 3 :
currents (including the dependence of carrier mobilities
upon
impurity
concentration
[ 7 ] and
electric
field
BA LA ‘v 3.
(9)
strength [8], [9]). From the particle currents and net
If n e assume for simplicity that B.4 varies as the sixth
generationrate(includinggenerationandrecombinapower of E,, we obtain
tion by field-dependent impact ionization
[3] and carrier concentration-dependent single-level recombination
Bn’ ‘v 6BA/EA
centers [ l o ] , [ l l ] ) ,it obtains the time derivative of the
or, using (9) and (4),
carrierconcentrationsfromthecontinuityequations.
From the time derivativesof the carrier concentrations,
B.4’ ‘V 6 X 3;’ V d c . d .
(10)
the program computes the carrier distributions an inEquation (3), theavalancheresonancerelation,
n-hen stant in time later, and repeats the cycle.
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
SCHARFETTER
AND
OSCILLATOR
DIODE
GUMMEL:
READ
SILICON
The diode operating characteristics are calculated
for
the diode imbedded in the circuit shoivn schematically
in Fig. 1(a), and (b). The variation of diode negative
conductance, susceptance, average voltage, and microwave power-generating efficiency withfrequency,dc
current density, and ac voltage amplitude was obtained
with the circuit shown in Fig. l(a). This method made
it possible tocontroldirectlythefrequencyandac
voltage amplitude in the calculation and was employed
to economize on computer time. Essentially the same
results were obtained with the circuitsholvn in Fig. l ( b ) ,
if theadmittance of the loadinductance L and load
conductance G Lvere properlychosen.Thecoupiing
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 inFig. 2 . The corresponding steady-state electric field versusdistanceforthisdiode
is shownin
Fig. 3. T o simplifythesignconventionon
field and
current density, thefigures representa n-v-n-9 structure
with the substrate on the left and the diffused junction
on the right. Note that the donor densityin the v region
is not negligible, but causes a continuous fall-off of electric field out to the substrateregion. This doping profile
evolved in trial runs to eliminate generation by impact
ionization asthecharge
pulse of electrons passed
through this region, while still allowing as large an average electric field in thedrift region as possible(see
Fig. 12 (d) I\-hich illustrates this effect).
Inagivenrunwith
fixed dccurrentdensity,ac
voltage amplitude,andfrequency,theprogram
\\-as
continued until the limit cycle was obtained. The terminal currentandvoltagewaveforms,during
a limit
cycle, were analyzedtoinvestigatethevariation
of
diode characteristics at various values
of oscillation amplitude. Thethreefundamentaldiodecharacteristics
investigated arenegativeconductance,susceptance,
and averagevoltage.Resultsare
sho\\-nin Figs. 4, 5 ,
and 6. Thediode’slarge-signalnegativeconductance
and susceptance values are effective values at the voltage generator frequency 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 required
to characterizecompletelythelarge-signalbehavior.
However, calculations made at other current densities
(100 to 1000 X/cmZ) indicate that the results are qualitatively similar to the detailed results presented
for a
current density of 200 A/cm2. The magnitude of negative conductance increases up to about1000 A/cm2, but
the efficiencies obtainedare less t h a n a t 200 A,/cm2.
Thediode’ssusceptance
is predominatelyduetothe
diode’sspace-chargecapacitance,andthereforedecreased with increasing current due to widening of the
space-charge width. The average voltage increases with
currentdensity,butbecause
of therapidincrease
of
ionization rates with electric field, the average voltage
67
decreases \\-it11 ac voltage amplitude \\-hen the average
current is held constant.Large-signalresultsatlarge
currentsare moredifficult toobtain \\-it11 thecircuit
shown in Fig. l(a)(voltagegenerator)
because parametric [12] effects occur
a t large dc current densities.
Thesusceptanceshown
in Fig. 5 is expressed asa
“capacitance” and plotted in Fig. 7. The value is only
slightly less than
the
depletion
layer
capacitance
(1.9 X lop9 F/cm*), n-ith thelargestdeviation
at the
lower frequencies. The ratio of susceptance to conductance Q is shown in Fig. 8, and the power-generating
efficiencies are plotted in Fig, 9.
The conductance and susceptance values are
sho1j.n
as an admittance plotin Fig. 10. Results for frequencies
between 6 and13.4 GHz andacvoltageamplitudes
between 0 and 38 voltsaresho\\-n
for adccurrent
densit>- of 200 -\.‘cm2. Alsoshon-n areconstant efficienc\- contours. The plot indicates,
for example, that
10 percent efficiency is obtained a t a frequency of 11.4
GHz \\-it11 a voltage amplitude of 30 volts, \\bile only
20 volts is required a t 8.4 GHz. I t \\-as found that the
efficiency increasesu-ithacvoltageuntiltheelectric
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 a t considerably lolver voltage amplitude for the
lower frequencies (less than 7 C H z ) , and made it very
difficult to obtain steady-state solutions
for thesefrequencies.Also,notetherapidincrease
in negative Q
(ratio of susceptance to conductance) \\-ith ac voltage
amplitude. This would cause the efficiency to turn over
and decrease with further increases
in ac voltage amplitude for deviceswithseriesleadandcontactresistance, since the diode shunt negative resistance’
is reduced by a factor of Q 2 in converting from a parallel to
a series equivalent circuit.
Detailed “snap shots” of the electric field, and hole
andelectronconcentrationasfunctions
of distance
during one cycle of steady-state oscillation are shon-n in
Fig. 11. The “snap shots” are shon-n a t approximately
one-fourthcycleintervalsinFig.
l l ( a ) through(d).
T h e figures are selected frames from a computer-made
movie [ 6 ] . .I phase plot of the terminal current I and
voltage 1- of the oscillation is included in the figures.
Points to note are the follotving.
1)Thegeneration of pulses of holes andelectrons
begins\\-here thevoltage is amaximum;one-fourth
cycle later, the charge pulses are fully formed and begin
drifting into their respective drift spaces.
2) T h e holes disappear quickl!. fromtheactiveregion while the electrons driftfor approximately one-half
cycle and constitute positive particle current while the
ac voltage is negative.
The calculationincludes all losses that originate between the
electricalcontacts, i.e., substrateandavalanche
space-chargeresistances.
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
68
IEEETRXKSA\CTIONS O S ELECTROSDE\.ICES,
I60
J-ASUARY 1969
13.4
12.4
140
3.60
3.2-
E
,o
VI
3
c
11.4
1201
2.8-
-9a 2.42.0lA
----0.4-
Fig. 3 .
Staticelectric field and particlecurrent profile versus distance i n dc stead?--state Read diode 7&1.
L
0
I
10
I
20
I
30
I
I
40
'50
IO
A
0
&
DISTANCEINMICRONS
A C VOLTAGE AMPLITUDE
Fig. 1. Diodenegativeconductance(mhos/cm*)
as a function of
acvoltageamplitude
for ~ a r i o u sfixed frequencies.Current:
density 200 .1/crn2.
30
40
AC VOLTAGEAMPLITUDE
Fig. 5 . Diodesusceptance(mhos/cm*)as
a function of acvoltage
amplitudeforvarious
fixed frequencies.Currentdensity:
200
A/cm2.
I
6
20
0
IO
I
I
I
20
30
40
A C VOLTAGE AMPLITUDE
I
50
I
0
Fig. 6 . Diode average voltage as a function of ac voltage amplitude
for various fixed frequencies. Current density: 200 A/cm*.
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
69
SCHARFETTER AND GUMMEL: SILICON READ DIODE OSCILLATOR
17.5I 850-
15.01,825-
8
W
g
a
12.5
-
>
1.800-
V
z
t
V
lg 10.0
a
a
a 1.775-
IL
IL
W
u
1,750-
-
7.5
5.02.5-
1.725-
-T
-
2
I
I
0
IO 30
I
I /
I
I
20
I
01
0
50
40
AC VOLTAGE A M P L I T U D E
IO
I
20
I
30
I
40
I
50
1
PC VOLTAGE A M P L I T U D E
Fig. 7. Diodesusceptance expressed asanequivalentcapacitanceFig.
(depletion
layer
capacitance=
1.9X
F/cm2)versusacvoltagevarious
amplitude for various fixed frequencies.Currentdensity:
200
A/cm2.
9.
Power-generatingefficiellcyversusacvoltageamplitude
fixed frequencies.
Current
density:
200 A/cm2.
160
I50
140
I30
I20
I IO
100
W
90
Q
80
x4c
3
2
0
70
60
50
40
30
20
01
I
0
IO
I
I
I
20
30
40
I
e
AC VOLTAGE A M P L I T U D E
Fig. 8. Diode Q (ratio of conductancetosusceptance)versusac
voltage amplitude for various fixed frequencies. Current density:
200 A/cm2.
I
-20
-15
IO
0
I
-10
-5
0
CONDUCTANCE
Fig. 10. Diodeadmittance(susceptanceversusconductance)asa
function of frequency and ac voltage amplitude, and resultant
efficiency indicated. Current density: 200 .4/cm2.
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
for
70
JEEE TRXKSXCTIONSONELECTRON
DEVICES, JANUARY 1969
-5.0
Y)
- 4.0
*
I
E
\
Y)
- 3.0 2
-a>
a1
A
- 2.0
u
a
+
u
0.4
W
J
W
- 1.0
Lb
0.
0
w
L
l
1
2
3
0.
I
4
5
6
7
8
3
IO
DISTANCE IN MICRONS
DISTANCE
MICRONS
IN
(C)
-5.0
-0.6 v,
w
a
U
-
0.4
- 4.0n0
*
.5
I
1
2
3
4
.
5
E
- 3.0
- 3.0
v,
->
-0
0
0
J
- 2.0
w
W
-
-20‘
9
0
-
K
+
u
+
u
W
- 1.0
I
I
5
6
7
8
*
-
E
-
0.
0
- 4.0:
-
/
E
-5.0
W
J
W
- 1.0
W
L0.
L0.
IO
9
DISTANCE IN MICRONS
J
9
DISTANCE
IN
IO
MICRONS
(b)
(a
Fig. 11. Solutions of hole and electron concentrations, electric field, and terminal current and voltage (values indicatedby
on phase plot) at various points in timefor the diodeoperating at a frequency of 12.1 GHz, current density of 200 A/cm*,
and 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 quite large and has
anappreciableswingintotheconventionalfor\\-ard
direction, n-hile the terminal voltage always remains in
the conventional reverse polarity.
5) The behavior is as predicted by Read, except that
forsilicon diodesanextendedavalancheregionisrequired 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 a t higher dc
current densities by trial and error choices for the load
con’ductance G and load inductance L. “Snap shots” of
the oscillation at a dc current density of
1000 A/cm2 are
shown in Fig. 12(a) through (d). The largest
efficiency
obtained for this bias (9 percent) occurred at an operating frequency of about 13.4 GHz. Kote that the carrier
concentration scales differby a factor of 5 between Figs.
11 and 12, as do the dc current densities. However, the
large-chargepulse, for the 1000 A/cm2case,moving
through the drift space results in a significant value of
peak field near the substrate just as the electrons approach this region. I f this peak field is too large, genera-
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
71
SCHAKFETTER AND GUhIMEL: SILICOK READ DIODE OSCILLATOR
-5.0
1
5.0
- 4.0
in
1;
.
0
-
P
-E
v)
4.0
*
-3.0 v)
5
E
- 3.0-
->
r
n
E
-1
- 2.0 w
a
V
2.0
-
Y
I
v)
r
3.0
-
w
U
a
V
2 0-
I-
.u
W
J
-1.0
0.
I 0
, 1
O2
3L
i
4
DISTANCE
5
IN
6
7
8
k 0.
9
IO
1.0
1
0
1
2
3
4
5
6
7
8
DISTANCE IN
MICRONS
MICRONS
(a)
(C)
-5.0
-4.0
m
-Iz*
.5
v)
-3.0 3
0
>
I
0
J
w
-2.0
U
+
u
W
J
- 1.0
W
DISTANCEINMICRONS
DISTANCE IN MICRONS
(a)
(b)
I;ig. 12. Solutions of hole and electron concentrations, electric field, and terminal current and voltage (values indicated b\.
0 on phase plot) at various points in time for the diode operating a t a frequency of 13.4 GHz, current density of 1000
A/cmZ, and efficiency of 9 percent.
tion byimpactionizationoccurs,andthesubsequent
drift of a hole charge pulse toward the p region would
constitutepositiveparticlecurrentnhilethevoltage
was alsopositive,averyunfavorablephaserelation.
Thiseffect was worse in an earlier doping profile (not
shon.n), and was eliminated by altering the
profile to
provide a continuous fall-off of electric field out to the
substrate (see Fig. 3).
The nearly ideal (classical Read [l 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 a n ac steady-state solution at 11.4 GHz and dc
current density of 200 A/cm2. The particle current was
1)
obtained by subtracting the displacement current from
the total current. At the largest
oscillation amplitude,
thediode“capacitance”changesparametrically[12]
and makes an accurate separation of the current componentsdifficult. The result shown inFig. 13 is for a
voltageamplitude
of about 17 volts.Holyever,the
waveforms for larger amplitudes are similar. The particle current is not a square wave during the half cycle
over which the voltage is negative, but has a hump resultingfromtheextracomponent
of particlecurrent
which flows until the holes that
were generated along
with the electrons are swept into the
p layer. For large
amplitudes, the hump becomesmorepronouncedand
the particle current bottoms [all carriers are swept out
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
72
IEEE TRASSACTIOSS O S ELECTRONDEVICES,JANUARY
1969
APPENDIX
SEMICONDUCTOR
DEVICEA S A L T S I S
COMPUTER
PROGRAM
1
INTRODUCTIOX
ThisAppendixconsists
of threesections.Section
I
describes the device physics included in the large-signal
calculationandoutlinesthesolutionprocedureswithout detail.SectionI1describesindetailthetimeadvancement techniques and the mathematical formulation employed in the computer calculation. Section I11
is a detailed discussion of the matrix inversion subroutine developed specifically for the time evolution computer program.
I. ~ I A T H E M A T I C A L~ I O D E L
A . Charge Transport Equations
0
TIME
Fig. 13.
Diode voltageandparticlecurrentversustime
cycle of steady-state oscillation.
for one
as shown in Fig. 11 ( a ) ] over a n appreciable part of the
cycle.
The resultsof 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,overanorder
of magnitudevariation in dc
current.Thefrequency
of operation,amplitudes
of
voltage and current oscillations, order of magnitude of
dc bias, and efficiency are in fairagreementwiththe
prediction of therelativelysimpledesigntheorypresented in Section 11.
The distribution and motion of carriers Ivithin a onedimensional semiconductor device structure can be obtainedbysolvingthreebasicequations:1)the
continuity equation forholes, 2 ) , the continuity equation
for electrons, and 3) Poisson’s equation:
1 d.T,
dP
-df= g - - - , q 8%
dl1
1 dJ,,
dt
q
_-g + -
->
dm
and
\\here
IV. COXCLVSIOSS
Operating characteristics of a silicon Read diode \\-ere
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-consistentsolutionswereobtained
for the
equationsdescribingcarriertransport,carriergeneration,andspace-chargebalancen-ithinthediode,and
n-hich simultaneously satisfied the boundary condition
imposed at the diode metallic contacts by the resonant
circuit.
Detailedsolutions were presented of the hole and
electron
concentrations,
electric
field, and
terminal
current and voltage at various pointsin time during ac
steady-state cycles of oscillation. The largest efficiency
obtained (18percent) is in good agreementwiththe
approximatecalculation,whichincludestheappreciablewidth of the avalanche region requiredfor silicon
devices.Efficientoperation
( 9 to 18 percent) was obtained over a range of dc current density (100 to 1000
A/cm2) and frequency (6.4 to 13.4 GHz).
Boundary conditions are imposed at the contacts by introducingtheappropriaterestrictionsin(13)through
(1 7 ) . For example, current boundary conditions
for a
p-n deviceareintroducedbyrequiringthat(17)
is
equal to the terminal current density
a t t h e n contact
and that (16) is equal to the terminal current density at
the ‘ p contact. \-oltage boundaryconditionsareimposed by requiring that the integral of E ( x , t ) over the
intervalbetweenthetwocontactsequalsthetotal
voltage. I n 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 quiescent zero
bias solution, i.e., the solution when
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
73
OR
E READ
SILICON
GUhlMEL:
SCHARFETTER
AND
In general, the functions g, p,, p n vary at each point
in the device according to the value of n , p , and E a t
that point.
B. Carrier Generation-Recombination Expressions
Thecarriergenerationterm
g iscomposed of two
components: 1) carriergenerationandrecombination
through defects, and 2 ) impact or avalanche ionization.
Hole electron generation and recombination through
defects
are
represented
by
a Shockley-Read-Hall
(SRH) single-levelmodelwhichcharacterizesdefects
with neutral and single-charge states. The generationrecombinationratethrough
single-level
a
center
is
given by
p n - ni2
gd
+ nd +
+ PI)
= --
rpo(n
Tno(P
(19)
Theimpactionizationratesarestronglydependent
upon the electric field intensity and the hole and electroncurrentdensities.Thesegenerationtermsare
given by
where an(E) and a,(E) are given by the relations
an(E) = 2.25 X lo7 exp (-3.2 X 106,/E)
(21)
a,(E) = 3.80 X lo6 exp (-1.75
(22)
X 106/E).
Ax
q
= -(P(X) - n(n.)
E
Holes
Electrons
s
480 4X1Ol6
s
A
F
P(NP,(M)
J,(M)
=
E(M)
J,(M)
=
12(-I- l)Pn(M)
E ( M )_ _ _ _ - ~
(1.0 - esp (-E(M)Ax))
(1.O - exp (- E ( M )Ax))
+
t ---
(1.0 - esp ( E ( M ) A $
Theseequationsprovidenumericallystableestimates
of the current density under all conditions. If the intermesh point voltage is small, these equations approach
thestandard differencerelations;whereaswhenthe
voltage change is large, they approach the drift current
density at either mesh point N or N+1.
+ (E/BI2
PO
(27)
-ITA (X))
where the Mth mesh point is located midway between
the major mesh points N + 1 and N . I t is customary to
employ next the standard difference approximations in
the current density expressions (16) and (17) and substitute these results in
( 2 5 ) and (26). However, it can
beshon-n thatthisprocedureleadstonumericalinstabilitywheneverthevoltagechangebetween
mesh
points
exceeds
2kT/q. Rather, (16) and (17) are
treated as differential equations in p and n with J,,, J,,
p,, pn, and E assumed constant between mesh points.
The solution of these differential equations then relates
J , and J , to the other variables:
C. Mobility Expressions
I t is necessary to include in the analysis the variations
of mobility ( p ) withtheelectric
field ( E ) andthe
ionized impurities density ( N o ) .
Thetheoreticalmobility
is approximatedbythe
following expression :
+ -I-o(A') -
R
81 6.1X103 1 . 6 2.5xlO4
1400 3X1Ol6 350 3.5X103 8 . 8 7.4X1O3.
(24)
D. Solution Procedures
Because of thenonlinearities in theequationsdescribing 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. Theequationsarethennormalizedtoreduce
redundant coefficient calculations, and standard difference approximations are used to approximate the spatial derivatives in Poisson'sequationandthecon-
E . Definition of Symbols
hole and electron concentrations, cm-3
electric field intensity, V/cm
electronic
charge, 1.602 X
coulombs
hole and
electron
current
densities,
A/cm2
hole and
electron
generation-recombination rates, carriers/'cm3.s
impactionizationgenerationrate
for
holes and electrons, carriers/cm3.s
hole and
electron
generation-recombinationratesthrough
a single-level
trappingcenter,carriers/cm3
.s
dielectric constant
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
I E E E TRANS.ACTIONS ON ELECTROXDEVICES,JA4NUARY
74
Boltzmann
constant,
1.38044X
J /OK
form,
b y the closed-form
expression
conventional
1969
(33). Written in more
absolute temperature, OK
total ionized impurityconcentration,
~ m - ~
(36)
concentration O f donor and acceptorEquation
(36) is thevectorequation we solveforeach
impurity atoms, c 1 r 3
increment t intime.
LYhen n e havesolved
it, \ve
and
PP,P n
c m 2 / v ' s incrementthevector
yo bythesolution
6y. To geta
low-doping 'le
PPO
i n better intuitive understanding of what we are doing, let
silicon, (480 c m 2 / V . s )
us move the term containing
31t in (36) on the rightlow-field low-dopingelectronmobilityhandside:
in silicon, (1400 cm2/V.s)
ionization coefficients
for
holes
and
(3 7)
electrons, carriers/carrier cm
hole and electron concentrations in the
conduction band \vhen the
Fermi level
-At titne 0 , the slope 5 isf(yo). At time t , i t is approxicoincidesLvith theenergy level of themately
fo+-Tf6y. Theratio
of the finitedifferences
single-level trap.
1I.
VECTOR-iVIATRIX F o R h f C L A T I o s
A . T i m eA d v a n c e m e n t
We represent the hole and electron concentrations by
the two-component vector y . The system of equations
then reads
i.e.,thetimederivative
of y equalssomenonlinear
functions f of y . W'e denote y a t t h e beginning of the
time interval under stud!- by y o , and denote the deviations of this value as time increases by 6 y ( t ) :
y ( 0 = yo
+ 6y(t).
(3 1)
(1 - M t ) S y = f*t.
For small deviations 6 y , we expand f ( y ) as
6 i = f(y0)
+ M6y.
(32)
is the matrix d f / d y and represents an integro-differentia1 operator. Kow, if 51,f, and 6y were ordinary
numbers, the solution to (32) would be
6y(t) = (eiMt - l)f/M
6 y ( t ) / t corresponds
slope
the to
ofstraight
the
line
0 and t . \That we aredoing - is
through
- thepoints
propagating the system according to the time derivative
a t some advanced time t a e t / 2 , rather than at the initial time. The term in 6y on the right-hand side of (37)
represents a feedback that gives the system stability.
I t is worth noting that the expansion off with respect
to y (the derivative matrix M ) is correct to first order,
but with regard to timeit is correct to second order in t.
The method of propagation (36) time evolves the system accurately for small time steps
t. However, if we
are near an equilibrium state and are interested not in
thetimeevolution,butonly
in theequilibriumsolut of infinitycan he taken n-ith full
tion,timesteps
feedback, i.e.,
B . The Derivative dlatrix
We define the vector y as having components of hole
concentration, electron concentration, and electricfield :
(33)
or
6y(t)
=f.t(l
+ M t / 2 + ( M t ) ? / 6 + . . . ).
(34)
and the right-hand side of the continuity equations as
Note that if the physical situation is one in which \ve the vector f:
are near a stable equilibrium state, then the eigenvalues
of the matrix M must be negative so that as timegoes to
(39)
infinity, 6y reaches a finite value.
For small but finite time steps, we introduce the inThe vector equation we solve for each time increment
verse of the first two terms of (34) to obtain our solution t (36) is
to (30) a s
L.
where Aft represents changes in fs at the terminals and
where we define the derivative matrix
Theoperator
A1 in thedenominatorroughlycorresponds to integration and as t goes to infinity, we reach
an asymptotic value within a factor of 2 of that given
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
75
ERREAD
SILICOX
GUMMEL:
SCHARFETTER
AND
The matrix elements of X are defined by the expanded
form of (40) shown below:
+ ( F L ) M*
y3.tr-l
+ (FR),?r*
=
X L ( 3 , -V)
* 6 H ( S - 1 ) + X L ( 4 , 17') * 6E('q7 - I )
+
+
(44)
X R ( 3 , N ) * 6H(-V f 1 ) + X R ( 4 , JY)* 6E(.Y + 1)
+
XM(3, S )
y a =
~ (RH)N (42)
where
(RH1.v
and
2(fo).v
* 6H(.\')
+ X M ( 4 , -I-*)6 E ( S )
+
I'ector equation (42) is solved by the computer program
FL(2, S ) * 6 F ( M - 1) FR(2, -Y)* 6 F ( M )
subroutine described in Section 111. T h e 2x2 matrices
= RH(2, -\.)
S L , X M , and X R are obtained by taking the partial
derivatives off.^ with respect to H@L andELE a t mesh
\\here N ranges from 1 to L R , and
points N - 1 , N , and N+1, respectively, while the two
component vectors FL and FR are obtained by taking
6 F ( M ) - 6F(M - 1) = (6H(.l-) - 6 E ( N ) )* D Y ( S )
the partial derivatives off.\. with respect to
FIELD at
where
mesh points IZI and 21- 1, respectivel!,.
111. ~ I A T R I ISVERSIO~X
SUBROUTIXE
Dl'(.\.)
=
(DX(M)
+ DX(M - 1))/2.
(45)
This section describesin detail the method for solving
Inputquantitiesandtheirrequiredrangeare
as
thediscretizedcontinuit).andspace-chargeequations
follo\vs :
for the quantities 6 H ( N ) ,6 E ( N ) ,and s F ( 3 I ) .The quanDAY.~r M = 0 , LR
tities 6H and 6E are the changesin the hole and electron
S = 2, LR
[XL],
concentrations, while 6 F is a normalized electric field;
6F is in units of carrier concentration. The boundar!.
[XM].h, S = 1 , L R
conditionshavebeenincludedintheequations,i.e.,
[ X R ] # S = 1, LM (LM = L R - 1)
the particle currents at the left contact enter into the
( R H ) N -1-= 1, LR
right-hand side R H ( 1 ) of the first continuity equation,
and the particle currents at the right contact enter into
-Y= 2 , L R
(FL)K
theright-handside
R H ( L R ) of thelastcontinuity
(FR)x
S = 1 , LM.
equation.Sincethetotalcurrent
atthecontacts
is
known, continuityequations 1 and L R containthree
Output quantities and their range are as follows:
houndary conditions. The fourth boundary condition is
6H.v = Y(1, S )
S = 1, L R
that the electricfield at the contactis zero. Since overall
space-chargeneutrality is builtintotheequations,
a
6E.v = Y ( 2 ,S )
.\- = 1 , LR
field of zero a t one contact forces a field of zero a t t h e
6F.11 = Y ( 3 , M )
M = 1, LM
other contact. The method
forsolving thediscretized
(note 6F(O) and 6 F ( L R ) are zero).
continuity equations will bedescribed for the case of
terminal-current boundar!. conditions.ThemodificaB . Dettrils of Inversion
tions to this method \\.hen boundary conditions are in
terms of terminal voltage
a mixture of voltage
\Ye represent the output quantities6H and 6E by the
- or
- and
current Ivill be described in Section 111-C of thisvector
y as
.Appendix.
-4. Input-Output
\Ye describe a computer subroutine which solves the
following three equations:
X L ( 1 ,S )
and similarly input quantities
RH,F L , FR as
* 6 H ( S - 1) + X L ( 2 , -T) * 6E(&17
- 1)
+
+
(43)
X R ( 1 , X) * 6H(,1' + 1) + X R ( 2 , X) * 6 E ( S + 1)
+
X M ( 1 , &4-) * 6H(L3-)
+ X M ( 2 , S ) * 6E(A7)
FL(1, S ) * 6F(M - 1)
+ FR(1, -I-*)6 F ( M )
The input quantities X L , X M ,
XR are represented by
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONSONELECTRONDEVICES,JANUARY
76
[XLI,
=
[XX LL (( 13 ,,
1,
X)
XL(2, S )
AT)
XL(4, X )
1969
Note that (48) can be written as
- ( U F ) L * 6F(L)
and (45) as
~ F (L 1) = ~ F ( L ) J ~ H ( L-) ~ E ( L )*} D Y ( L ) ; (53)
therefore, once we have [ U M ] , ( U F ) , and ( 2 ) a t mesh
points 1 to L R , we can n-ork backwards using (52) and
(53) to find
and ~ F L - Ifor L = LR .
1.
The computation procedure is as follows.
1) Compute starting values from (51).
2) Computevaluesattheremaining
mesh points
(
2
,
L
R
)
using
(50).
We assume that \ye can remove the ( Y ) . Y - ~ term in (47)
3) Use ( 5 2 ) and(53),starting
at mesh points L R
and write the resulting equation at mesh point N-1 as
(note ( y ) L ~ + 1and 6F(LR) are both zero) and working
[CMIN-l * ( Y ) N - l
[URIN-l * (y1.v
(48) through mesh point 1 to obtain the values for 6H, 6E,
and 6F.
(UF)N-l * 6 F ( M - 1) = @)x-1.
where
+
+
Recursion relationships for [ V M J , [ UR],( U F ) , and
(2)are obtained as follows.
Equations (47) and (48) arecombinedtoeliminate
(y).v-l terms as
{ [ x M ]-. [L'FL].v
~
- [ ~ ] *x [ ~ ' ~ ] . v -*1(y1.v
]
[ X R ] s* ( ~ 1 ~ 1 (~ F 1N M (FL).,r) *
- p 1 . v * (CF),V-l * 6 F ( M - 1)
= ( R H ) . y - T2v* (2)s-l
p1.v = [XLIX * [ z - M ] N I 1 .
+
+
+
the desired boundary conditions could be incorporated
intotheinput
coefficients. The terminal currents enteredintotheright-handside
(RH),v at the contacts
(
N
=
1)
and
(
N
=
L
R
)
.
RIore
general
boundary condifiF(m)
tionscanbeincorporatedbyusinganeconomical
(49)
superposition process.
Withthetimeadvancement
processdescribed
in
Section 11, we n-ant to solve the equations
Theterm in 6F(m- 1) can be eliminatedin (49) by
(45), and therefore by comparing (49)Irith(48)evaluated at mesh point N , n-e get:
[ m ] , v = [XRIY
[IW].v
.)I:(
(Z'7F)1=
=
[XM]s
- [L-FL],v-
C. General Boundary Conditions
In the discussion thus far, i t has been assumed t h a t
[T]X * [z'R].v-l
[f - "21
* (Y) =f,
(54)
where t is thestep intimeand
M is thederivative
matrix. If the terminal currents are changing with time,
then f is fo, the right-hand side of the continuity equation at t =0, plusone-half of thechangeinterminal
current (the extra term will appear in only two equations, mesh points l and L R ) . Let us multiply (54) by
2 and write it as
where A j represents the change in terminal currents in
the time t. Equation ( 5 5 ) is identical to (46) and could
be solved by the method discussed previously in Section
111-A of this Appendixif the changein terminal currents
(Af terms) are known. What
is alwaysknown a t t h e
terminals is one of three possibilities:
1) the terminal current as a function
of time is an
independent variable;
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.
77
R
E READ
SILICON
GU1\’lMEL:
SCHXRFETTER
AND
2) the terminal voltage as a function of timeisan
independent variable; or
3) an equation relating terminal current to terminal
voltage is known.
The general solution to (55) for
Af arbitrary can be
obtained numerically using superposition. Letus assume
that our final solution will be the sum of two solutions:
(Y) = (Yo)
+ (YJ.
(36)
These solutions have the following interpretation:
(yo) represents the solution for Af = 0 , that is, for no
change in terminal current over the time step t ;
(yc) representstheadditionalsolution
in terminal current.
T o see this more clearly, let
for a change
u s n-rite ( 5 5 ) as
[ M ] N ( y ) = (RHOIN
+ (RHc).V
(ji)
where the quantity 2 l t + d i in (55) has been n-ritten as
[ I l d ] .in
~ ( S i ) and 2fo as (RHo).v and Af a s (RHC).y.
Equation (5i) has (RHc).vzero everywhere, except a t
mesh points 1 and LR.Therefore, u-e solve ( S i ) using
the method described in Section I I I - L lof this -Appendix
n-ith (RH).y = (RHO).,.. This yields solutions (yo).,-. S e x t ,
we solve ( 5 7 ) with (RH).\T = (RHc).\-;this yields solution
(yc).,-, Since Ive donotknowthechangeinterminal
current A J , we obtain solutions (yc).v for a unit change
in terminal current. Lye then integrate the electric field
component of (yc).y toobtainthechange
in terminal
voltage per change in terminal current Av,. If the integral of theelectric field component of (yo).\. is called
AI’,, then the change in terminal voltage A I 7 is related
to the change in terminal current A J b\AV = AVO
+ Avc * AJ.
(58)
REFERENCES
[l] \V. T. Read, “A proposedhigh-frequencynegative-resistance
diode,” Bell Sys. Tech. J . , vol. 37, pp. 401-466, March 1958.
H.K.Gummeland
D. L.Scharfetter.”Avalanche
region of
IAIPXTTdiodes,” Bell Sys. Tech. J . , vol. 45,pp. 1797-1827,
December 1966.
C. A. Lee, R . X. Logan, I<. L. Batdorf, J. J . Kleimack, and ii-.
iViegmann, “Ionization rates of holes and electrons in silicon,”
Phys. Rev., vol. 137, pp. i\761-X773, May 1964.
M. Gilden and 1’1.B. Hines,“Electronictuning effects i n the
readmicrowaveavalanchediode,”
I E E E T r a n s . ElectronDevices, vol. ED-13, pp. 169-175, January 1966.
T. Misawa,“Negativeresistance in p n junctionsunder avnlanchebreakdownconditlons,parts
I and11,” I E E ET v a n s .
Electron Devices, vol. ED-13, pp, 137-151, January 1966.
D. L. Scharfetter and H. K. Gummel, “Design of Readdiode
oscillators,”presented at the 1966 IEEE Solid-stateDevice
Research,Conf.,paper I11 b-4.
J. C. Irvln, “Resistivity of bulk silicon and of diffused layers in
silicon,” Bell Sys. Tech. J., vol. 41, pp. 387-410, hlarch 1962.
.A. C. Prior, ..‘‘Field dependence of carrier mobility in sllicon and
germanium,' J . Phys. Chenz. Solids, vol. 1 2 , pp. 175-180,
January 1960.
T. E. Seidel and D. I,. Scharfetter, “Dependenceof hol: velocity
upon electric field ;tnd hole density for P-type silicon, J . Phys.
Chem. Solids, vol. 28, pp. 2563-2574, 1967.
\V. Shockley and I\’.
T. Read, Jr., “Statistics of the recombination of hole and electron,’’ Phys.Rev., vol.87,pp.
835-842,
September 1952.
R. X. Hall, “Electron-hole recombination i n germanium,” Phys.
Rev., vol. 87, p. 387, July 1952.
B. C.DeLoach. Tr.. andR. L. Tohnston.“.\valanche
transittime
microwaveoscillators
an& amplifiers,” I E E ET r a n s .
Electron Devices, vol. ED-13, pp. 181-186, January 1966.
Authorized licensed use limited to: University of North Carolina at Chapel Hill. Downloaded on April 09,2010 at 21:39:33 UTC from IEEE Xplore. Restrictions apply.