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Modelling of VHF and microwave power transistors operating in quasi-saturation
Solid-State Electronics VoL 25, No. 8, pp. 723-731, 1982 Printed in Great Britain.
0038-1101[821080723-09503.0010 © 1982Pergamon Press Ltd.
MODELLING OF VHF AND MICROWAVE POWER TRANSISTORS OPERATING IN QUASI-SATURATION A. W. ALDEN CommunicationsResearch Centre, Ottawa, Canada and A. R. BOOTHROYD Department of Electronics, Carleton University, Ottawa, Canada (Received 25 March 1981;in revisedform 22 October 1981)
modellingof VHF and microwavepower transistors operating under quasi-saturation conditions is treated. Physical effects in the collector region are considered to be dominant in these devices under such circumstances, and a representation of the collectorinvolvinga variable-widthcharge-storageregion, modelled in two-lump form, is developed in order to characterize device behaviour. A systematicprocedure is presented for the evaluation of the parameters of the collector region model. The model has demonstrated its ability to describe device characteristics and performance under d.c., step transient and large signal sinusoidal drive conditions; typical results are presented. Abstract--The
1. INTRODUCTION
This paper presents a device model for the characterization of d.c. and dynamic large signal performance of VHF and microwave power transistors when operating in the quasi-saturation mode. Transistor behaviour in quasi-saturation has been considered by a number of authors[l-7]. Kirk[l] described the lowering of the reverse-bias voltage of the collector junction due to voltage drop across the ohmic region of the collector, with the junction becoming forward biased at the higher current levels. Beale and Slatter[3], and Pals and de Graaff[4], amongst others, considered injection of minority carriers into the collector under these conditions and developed physical models for d.c. operation. More recently, it has been recognized [8, 9] that a singlelump model of the collector region is unable to represent adequately such injected minority carrier charge under large-signal transient conditions; the models developed, however, do not specifically model the storage of charge in a finite region of the collector. In the present work, a lumped model of the collector region is developed which corresponds with the physical conditions that exist in quasi-saturation, and which characterizes behaviour of the device for d.c. and dynamic operation. Methods are given for determining the defining parameters of this model by measurement. The devices under study are illustrated in Fig. 1 devices of overlay type with interdigitated and mesh emitter structures. These device structures are designed to minimize edge effects around the emitter, and can reasonably be treated as a parallel combination of many, essentially one-dimensional, intrinsic transistors for purposes of analysis. The definition of an "intrinsic" device, as defined here, is indicated in Fig. 1. Specific transistor types considered, for which results are reported, are 2N1038A and 2N3375, of 1.1 GHz and 400 MHz .fT respectively.
For the types of transistor under consideration, detailed computer simulation[6, 7, 10, 11] has shown that under conditions of quasi-saturation a narrow space charge region exists in the vicinity of the base-collector metallurgical junction, corresponding to a forward-biased condition[l, 3, 4], and that the N - epitaxial collector may be divided into two distinct regions, as indicated in Fig. 2: (a) an approximately neutral injection region adjacent to the collector-base junction, over most of which conditions of high-level injection exist; Co) an ohmic region, contiguous with (a) and extending to the N + collector region, operating under conditions of negligible conductivity modulation. The excess minority carrier charge stored in (a) is generally much larger than that stored in the base, and is responsible for major dynamic effects. This charge is constrained within the injection region (a) by the electric field associated with the potential drop across the ohmic region (b) due to majority carrier flow, which is responsible for the resultant forward bias applied across the collector-base junction. The potential drop across the injection region, caused by the electric field distribution associated with high level injection, is very small. For given collector-base voltage, as the collector current is increased, corresponding to increased emitter-base forward bias, the injection region (a) extends farther towards the N + collector contact, and the ohmic region (b) shrinks in width. The two extremes of the quasisaturation regime are where the N - collector region is entirely "injection" on the one hand, and entirely "ohmic" on the other. The modelling problem is thus to represent a neutral charge storage region of variable width, depending upon bias conditions, coupled to an adjacent ohmic region. In the following sections, the collector region model is developed from d.c. relationships in quasi-saturation, 723
A. W. ALDENand A. R. BOOTHROYD
724 [.
AREA AC
~ABEA ~ A E = BASE CONTACT DIFFUSION I EMITTER
P
with the current transported through the region. On the basis of one-dimensional current flow, throughout the collector region
] I I
] I
I
I'
J = Jr, + Jp = constant
SE
I
I
I [ I
I N EPITAXiAL COLLECTOR , I ' I II
I"
EXTRINSIC DEVICE
I
since both 3", and Jp must be continuous and Jp is negligible in the ohmic region. The injection region, essentially neutral, contains equal excess charges of electrons and holes resulting from the forward bias of the collector-base junction, but flow through this region is by electron transport only.
LI
Fig. 1. Cross section of interdigitated power transistor structure.
2.1 Injection region
In the injection region, 0 ~
COLLECTOR/BASE JUNCTION REGION
"t
°Hct
REGION R
REGION
IB
(2)
Xi
&
+ ~
---'wv--
C
z ==
Xe
_ kT l dp q p dx
kT d(ln p) q dx
(3)
Upon substitution into the expression for Jr,, with the high level and neutrality conditions (p=n>>NepO assumed, the electron current density is given by the familiar high level expression
.x
J~ = q( 2/9, ~dn = - IJAE
Fig. 2. Definition of injection and ohmic regions of epitaxial collector. with a two-lump representation of charge storage in the injection region. Methods of determining the parameters of this model from appropriate measurements are presented, and its d.c. and large signal dynamic performance are illustrated.
(4)
where AE is the emitter area. The associated distribution of excess hole and electron density is triangular, of slope (JJ2D,,q), as illustrated in Fig. 3. The expression of eqn (4), and the triangular distributions of excess stored charge, apply over most of the injection region, losing validity close to x = x~. From eqn (4), the corresponding hole density p(0) at the junction boundary x = 0 is
2. d.c. MODEL OF COLLECTOR REGION IN QUASI-SATURATION
The device region under consideration is here referred to as the "intrinsic collector" as illustrated in Fig. 1. This region is assumed to be part of a one-dimensional structure, as shown in Fig. 2, of cross sectional area AE equal to that of the emitter and extending through emitter, base and collector. The N - collector region, of uniform donor density Nop~ and total width x,, is regarded as divided into an injection region adjacent to the junction, and an ohmic region. In the injection region, high level injection is assumed, so that p,n >>N~pi. In the ohmic region, negligible conductivity modulation is assumed with n = Ncp~. The inconsistency of these assumptions at the boundary x = x~ between the two regions implies a gradual transition from high to low injection conditions in the vicinity of x~; i.e. between p>>N~p~ in the injection region and p "~Nepi in the ohmic region. A reasonable definition of the boundary x = xl, adopted here, is thus
p(x~) =
Noel.
p(O) = Lx,/2D, qA~.
(5)
The hole density of eqn (5) has to be in agreement with the value given by the injection law of the collector-base junction. On the basis of low-level injection conditions in the base, ni2 p(0) = ~ exp (q Vjc/kT)
(6)
where Vjc is the voltage applied to the base side with
=
pl,n~
Jn
-Dnq
(1) I
The starting point in the analysis and d.c. modelling of the collector region is the assumption that recombination current is suflicently small to be neglected in comparison
x,
Fig. 3. Carrier density distributions in the collector injection region.
Modelling of VHF and microwave power transistors operating in quasi-saturation respect to the collector side of the junction. Simulation studies[10, 11] have shown that over a wide range of quasi-saturation conditions, the collector injection region can be in high injection while the base region operates at low level, so that eqn (6) is applicable. Otherwise, this relationship needs to be modified to allow for high level effects in the base. Owing to the presence of the (negative) electric field caused by carrier injection, a potential V(xD is developed at x~ with respect to the junction region boundary x = 0, where
f ; ' ~ kT 1 dp l
V(xD
i
=
1--q-p TxxJdx
kT q In [[ pp(O) -~)j
(7)
Using the junction law, eqn (6) together with the definition of eqn (1) for the location of the boundary x = x~ between the injection and ohmic regions, we obtain with Beale et al. [12], V(x,) = V , c - Vo
(8)
where
v°=k-~-Tq ln(N-m'}2"\n~ /
(2)
With the equilibrium condition as reference, the potential of the boundary at x = x~ with respect to the base contact, neglecting base resistance, is thus - Vo. 2.2 Ohmic region An externally applied bias voltage Vca results in a voltage VcB+ Vo across the ohmic region. Thus,t assuming negligible conductivity modulation, V~,,+ I,'o V ' ~ + Vo Ic - x, - x-~ " qlz"NcpiAE = r~o(1 - x~x,)
is required that
2 D n'(O). A~ q " x~
VcB + Vo r~o(1-x.Jx,)
(11)
where n'(0) is the excess electron density at the collector junction boundary. 3. DYNAMIC CHARGE STORAGE MODEL OF COLLECTOR
Under conditions of quasi-saturation operation, charge storage in the injection region of the collector exerts a major influence on the dynamic behaviour of the device. The charge stored in this region can be considerably greater than that stored in the base. Adequate modelling of the dynamic charging effects associated with this region is therefore important, and should characterize with sufficient accuracy the distribution charge flow in a storage region of time-dependent width, with a minimum of model complexity. As in the comparable situation of charge storage in a P N junction diode, a two-lump charge storage representation of the region is chosen as being the simplest form of model which can meet these requirements. Conditions in the two charge storage lumps are defined by the boundary conditions at x = 0 and x = x, together with the requirement of continuity of current flow at their common boundary. 3.1 The basic two-lump model As indicated in Fig. 4, the injection region is divided into two storage lumps, of widths x~, x2 containing excess majority carrier charges ( - ql), ( - qz), where the ratio
K = xl/x2
(12)
can be chosen arbitrarily. These stored charges are related to the average excess majority carrier densities nl' -- ql/qAExl n2 - q2/qAex2J
(10)
I
~
where rco is the resistance of the full ohmic region when x~ = 0. For given conditions of operation VcB, Ic in quasi-saturation, eqn (10) defines the position of the boundary x = xi. The current expression eqn (4) in the injection region then specifies the hole injection level p(0) at the collector junction boundary, as in eqn (5), and hence the forward bias voltage of the collector-base junction by eqn (6). The same value of lc needs to be specified by the base region model, with an appropriate choice of applied VsE or of base current. The modelling problem is that of representing the current flow and associated charge storage properties of a collector region of variable width x, where xi is such that eqns (4) and (10), for the collector current flowing through the injection and ohmic regions respectively, are simultaneously satisfied. Thus, for the d.c. steady state, it
725
1 I
Nep i
(13)
•
I I
I
X1
Xi
× Xe
(a)
n
i
I Xe
X
XI÷X 2 (b)
tBase resistance rw is ignored in this discussion; strictly, with base resistance included, Vc8 should be replaced by VcB,.
2
Fig. 4. Definition of 2-lump charge storage model of the collector injection region.
726
A.W. ALDENand A. R. BOOTHROYD
respectively for the two lumps. For given charges and value of x. the current I, flowing from lump 2 to lump 1 (corresponding to electron flow in the opposite direction), and the current/2 flowing from the ohmic collector region into lump 2, may be expressed by difference equations in terms of n'~ and n~', these being regarded as equivalent electron densities at the centre point of the respective lumps. Thus, for high-level transport of majority carriers, neglecting recombination effects, L
-
q A E ( n l - n~). 2D. (x, +x2)/2
(14)
qAnn~. 2D, x2/2
(15)
and
12=
In terms of the corresponding stored charges, = (q,-Kq2)/¢3
(16)
= q2/¢4
(17)
with inclusion of the charging currents q~, q2 of the storage lumps and, as a second order effect, recombination currents ql/rl, qJ'r2 (where ¢~, ~'2">r3, ~'4). The current C~(~cB, which represents charging via the ohmic region boundary, is treated in Section 3.2. 3.2 Dynamic boundary conditions--region width modulation Under dynamic conditions, the currents associated with the stored charges q~, q2 are as discussed above and represented in Fig. 5. Thus, the current flowing from Jump 1 of the collector model into the base is (I, + qt/~ + q~), and must be consistent with the current value given by the base region model for the applied junction voltages. However, in addition to the currents so far considered, which correspond to a specified region width x, it is necessary to represent the charging current flowing at the ohmic region boundary of the injection region (x = x~) due to variation of this width. For the toal stored charge q~( = q~ + q2) in the injection region, we may write dqc
where the "transit time" parameters r3, ~'4 are r3 = r3o(XJXe) 2
(18)
x, 2 K r3o = 4/), " I + K
(19)
r , = r,o(XJX+) 2
(20)
with
and
I,, = ~
z+o K(1 + K)'
(21)
Figure 4 illustrates these relationships for the d.c. steady state, in absence of recombination of excess charge; in this case, evidently I, = I2. Figure 5 shows the model representation of the collector injection region under general dynamic conditions,
[ .......
~q" I
vsc........
~ _ (ql_Kq2)/.t,3
Ic I ]
q~
ql/rl q2
HOLE FLOW INJECTIONREGION JUNCTION REGION
d Vcs = c d Vc~ dt ~,c dt
C,c - - q" n'(O)Ae . dxi 2 d Vcn"
C
rc [ BASE
(22)
(23)
This charging current is included in the model of Fig. 5. The capacitance C,+c, evaluated from the static triangular charge distribution in the injection region (with recombination effects neglected) is
C.C dVcB dt ELECTRON FLO~W
aq¢ ax~ t q" ax~ at
where the second term is the charging current entering the injection region through the boundary at x = x+ due to change in region width alone (constant boundary conditions at x = 0). In the model, this charging current flows into lump 2, and is evaluated on the basis of quasi-static conditions. Accordingly, we define a charging current I,+c in terms of the change of charge dqc resulting under static conditions with the collector junction voltage constant and the injection region width changed by dx+ due to a change d VcB in collector voltage, in the same manner as in the modelling of base width modulation, and write
with T3O
aqc
"~'-=-"~'lxi
q /r
I I I I I I OHMIC 19
REGION
Fig. 5. Lumped model of the collector for quasi-saturation operation.
(24)
Modelling of VHF and microwavepower transistors operating in quasi-saturation From eqn (11), dx~ d Vc~
-x~ 2
2qAED, n'(O)r~o
(25)
giving
x~2 =C c~c = 4D.rco
( x~]z ,,CO~xS/
(26)
727
specifications, Vo, and r~o are known; otherwise, separate determination of these parameters is necessary. It is shown below that, with K specified, the model parameters can be deduced almost entirely from d.c. characteristics of the device, supplemented by an observation of transient behaviour when the device is switched between different quasi-saturation states. Typical results are presented for type 2NI038A and 2N3375devices. 4.1 Parameter determination from d.c, measurements
where
1 + K. r3_..~o C"~°=
K
(27)
rco
3.3 Complete device model A complete device model incorporating the above modelling of the "intrinsic collector" can be developed along conventional lines. Neglecting fringing effects, the remainder of the collector (outside the intrinsic region) may be regarded as being purely ohmic with the collector junction reverse-biased. This region has collector series resistance t__
A~
rc - r~o • A c - A ~ '
(28)
By interpretation of the d.c. characteristics of the transistor in terms of the relationships presented above, all the parameters needed for d.c. characterization of the collector model can be obtained. Three sources of data are utilized relating to the following features of the device characteristics in quasi-saturation: saturation] quasi-saturation boundary ( I c - V c e locus), active/quasisaturation boundary, common-emitter current gain in quasi-saturation. The saturation/quasi-saturation boundary locus across the common-emitter collector characteristics can be estimated, as indicated in Fig. 6 (e.g. points a, b, c). Along this locus, since rc = 0 and the injection region occupies the entire collector epitaxial region, Vo = VcE + VEB + IBrbb' = Vca'.
where A c is the total collector junction area and area ( A c - AE) carries only collector junction charging current. The base and emitter regions of the transistor are suitably modelled for example by a Gummel-Poon transport model in conjunction with an extrinsic base resistance. 4. DETERMINATION OF THE PARAMETERS OF THE MODEL
For the representation of quasi-saturation operation of the transistor, an intrinsic collector region model has been presented which is specified by the following parameters: Parameter
Definition
Vo(or Nepi)
eqn (9) eqn (10) eqn (12) eqns (19) and (21) Fig. 5 eqns (26) and 27)
rco(Orx,) K ¢3o,~4o r~, ~'2 C~co
These parameters need to be determined for a given device, in addition to those of the rest of the complete device model. We shall here be concerned with procedures for determining the above parameters of the intrinsic collector model, the remaining model parameters being assumed evaluated by familiar methods of measurement. Of the parameters listed, the "lumping parameter" K is arbitrary. The choice K = 1 is made here, corresponding to the injection region of the collector divided into two lumps of equal width. If the epitaxial region parameters Nepi, xe are known from device design
(29)
Consistent values for Vo have been obtained for samples of both types of device studied (2NI038A and 2N3375) over a range of points along the boundary locus. For 2N3375 devices, good agreement has also been obtained with Vo values calculated from eqn (9) for a known N,p~ value of 1.6× 10~cm-3: e. g. at 25°C, Vo=-0.60 from eqn (9) as compared with an average value of - 0.59 from the d.c. characteristics, with a spread of values of -+5% in the latter for the range of locus points taken. Once Vo is found, the collector resistance for any bias point in quasi-saturation follows from eqn (10), viz. rc
v¢~, + Vo - = rco(l - x~x,). Ic
(30)
The boundary locus between the active and quasisaturation regions (shown as 1, 2, 3 on the collector characteristics) of Fig. 6 can be used in order to determine rco from eqn (30) since along this locus the collector epitaxial region is entirely "ohmic" and xi =0. Results for rco consistent to within 10% accuracy have been obtained for the 2N1038A and 2N3375 devices using points along this locus, giving average values of 5.3 and 8.51"1, respectively. The result for the 2N3375 compares reasonably with 9fl determined by Anderson et al.[13] using an admittance bridge method, and 11.6[1 calculated from available design data for this type of device (emitter area = (12.7 g,m)2x 160, x¢ = 10•m, Nepi = i.6 x 10'5cm-3). With rco known, the ratio xdx, follows from eqn (30) for any bias point in quasi-saturation. At this stage in the parameter evaluation process, we are left with the characteristic times ~'3o, ~'~o, r~, ¢, and
A. W. ALDENand A. R. BOOTXROYD
728
160--
giving
I
140~
Ic = Ic(1 + llflF) Is ~o. Is - IciEr
/
2N3375
120--
100--
_o
where /3F is regarded as known (e.g. measured at the same lc value for Vcs, = 0). This current ratio can also be expressed in terms of the model parameterg. Referring to Fig. 5, we have
2
SATURATION
IBm2 mA
80--
(33)
// ~/~ATU"ATION/-'/4~
i-QUASI-/ /
Ic = q2]7-4
60-and with r, = 7-2,
40-20-0
In coil= ql + q2
TI
f// / I
I
I
I
I
I
I
02
04
0.6
0 8
1.0
1 2
1.4
Now, for d.c. current continuity within the model
VCE VOLTS
Fig. 6. Collector characteristics of 2N3375 transistor, defining operating regions (measured). the capacitance C,~o yet to be found. It should be noted that for the modelling of d.c. relationships, only the ratios of the "time parameters" are required. Thus, making the reasonable assumption that the recombination lifetimes r~, r2 for the two charge storage lumps are equal, only the values of the ratios r4o/7-3o, rl]7-3o need to be known fro purposes of d.c. characterization. However, from the defining relationships of the twolump collector charge storage model, with K chosen as unity, we have (see eqn 21).
so that q_2+ K(1 + K) + K + 2
q2
7"1
and Ic rt K ( K + 1) Is co----~i- 7-3 {1 + K ( K + 2) + r317-,}"
Using the condition rl "> 7-3,with K = 1, we obtain
/C
IB~o. 1
1
Z4oh'ao K ( K + 1 ) - 2
(31)
so that only r,/7-3o remains to be evaluated. The common emitter current gain properties in quasisaturation provide a source of data from which this remaining d.c. parameter can be deduced. In quasisaturation, the current can be written IB = IB ~
+ Io col,
(32)
where the second term originates in the collector injection region. The current gain is reduced by only a modest factor compared with active region operation. Thus, since for the type of device under consideration Neo~ is small compared with the level of base doping, it is reasonable to expect the base current component originating from reverse injection into the base to be of significantly smaller magnitude than either IB b,~ or IB c,,,. Hence, with reference to Fig. 5, 1. h..o = (1c + 1~ co.)//~F and I . ~o,I = I . - ( I c + I~
co.)l~
(34)
7-1
2r{
(35)
The parameter ratio 'rl/'T 3 c a n therefore be found for any point in quasi-saturation by utilizing eqns (33) and (35). Having previously determined rco, xJx, is also known for this bias point from eqn (24); the ratio 7-~/Z3ois known from eqn (18). The process of d.c. characterization is complete at this stage. Figure 7 shows typical agreement obtained between measured and computed collector characteristics for a 2N3375 device, using the model relationships and the parameter values listed in Table 1. The parameter values obtained for the 2N1038A device are given in Table 2. 4.2 Parameter determination [rom collector transient response The parameter determination process is completed by the evaluation of the "transit time" parameter 7-3o,which characterizes current transport, through the carrier injection region of the collector. The width modulation capacitance parameter C, co, related to 7-30by eqn (27) is then also known. Thus, the dynamic behaviour of the collector region in the quasi-saturation mode is dependent upon this single parameter as far as time scale is concerned, all other relationships having been established by d.c. measurements.
Modellingof VHF and microwavepower transistors operatingin quasi-saturation
160-
729
MEASURED 136-
2N 3375 J
""
i
134IcmA (COMPUTED) 132--
140X i/Xe = 0
120--
~#~,,-,~
130--
//r3°=6° ns j r / r 3 0 = 6 ns .
BOUNDARY ~ , " "
100-
.
.
.
.
.
.
.
.
.
.
IC-2 mA 128(MEASURED) - 126 --
. - " " "-
E 80-
124
I o
I 1 t ,,sec
I 2
60-
///f-
40-
Fig. 8. Collector current response to a collector voltage step while in quasi-saturation.Comparisonof response of the model with measurementfor type 2N3375device.
........
l/
EABU EO
20-
['
.........
?gg2E RA TEnB TABLE I1
I 0.2
0
I 0.4
I 0.6
I 0.8
I 1.0
I 1.2
I 1.4
VCE VOLTS
Fig. 7. Comparison of collector characteristics computed from the model with those measured, for type 2N3375 device sample.
Table 1. Collectormodelparameter values: 2N3375 P a r e
Value
v 0
-0.6 volt
rco
8.5~
K
1.0
x40/x30
0.5
Xl/X30
78.6
x 2/x30
78.6
•t
30
4.3aS
Table 2. Collectormodelparameter values: 2N1038A parmmter
Value
V0
-0.6 ~olt
•co
5.3fl
g
1.0
x4~x30
0.5
Xl/X30
340
x~x30
340
x30
86pc
The value of r3o may be deduced from the transient response of collector current when the transistor is "switched" from one quasi-saturation state to another by applying a step voltage drive to the collector, with the transistor in common-emitter configuration and base current maintained constant. When this is done, the transistor undergoes a transition between states corresponding to different values of injection region width
x, or of charge storage in the collector. The method used is to match the collector current response given by a computer implementation of the transistor model to the response observed in the test on the device, by adjusting the one remaining "free" parameter ~'3oof the model for the best agreement. Figure 8 illustrates the typical comparison of measured and computed collector current response waveforms, with several values of r3o, for a device sample of type 2N3375. With 18 = 3mA, a step change of Vc~ from 0.80 to 0.85 V is applied, taking the transistor to a state less deeply in quasi-saturation. Because of the relatively small changes in Vcr and Ic, the response waveforms of Fig. 8 are aligned to eliminate the initial discrepancy in lc between the model and the actual device. The best choice of r3o was judged to be 4.3 nS, which is in good agreement with the value of 4.17 nS calculated from eqn (19) with the design value of 10 #m for x,. The values of the parameters rso, ~',,o, r,, r2, and C, co presented in Table 1 for the type 2N3375 device all followed from the determination of r3o by the transient test method, taken in conjunction with the parameters previously obtained from d.c. measurements (as did for those for the 2N1038A device in Table 2). The transient test method can also be used to determine the value of collector resistance re. Since the collector injection region width x~ cannot change instantaneously on application of a step change A Vcs, there is an initial step change A VcB[rc of collector current as shown in Fig. 8. Values of r~ obtained by this transient test method have been found to agree with those from d.c. measurements as described in Section 4.1. Use of the transient test method is advantageous where the boundary between the saturation and quasi-saturation regions is ill-defined on the d.c. characteristics. Under these circumstances, determination of Vo from this boundary (see eqn 29) is unreliable and an alternative procedure is to use the d.c. relationship of eqn (30), with rc determined from the transient test, to obtain the value of Vo. 5. MODEL IMPLEMENTATION AND RESULTS
5.1 Implementation For purposes of d.c. or large-signal dynamic analysis, the collector region model defined by the relationships presented above can be best implemented as a computer
A. W. ALDENand A. R. BOOTHROYD
730
were presented in Section 4 in connection with model parameter determination. Figure 7 shows examples of the degree of matching obtained between measured characteristics and those given by the model. The ability of the model to represent dynamic behaviour of the collector region is indicated by the results shown in Fig. 8 for collector current response to a step change in VcB. The close agreement with the measured response gives confirmation of the adequacy of representation by the model of the charging conditions at the boundaries of the collector injection region and the consequent change of its width with time; the initial choice of a two-lump representation of this region, and the application of charge control relationships appear to be well justified. In the implementation of the model used to obtain the above results (and those discussed later in this section), certain simplifying approximations were made, justified by conditions referred to in earlier sections, namely: (i) forward injection dominant in the base region (reverse transport current neglected); (ii) depletion layer charging currents neglected in comparison with injection charging currents owing to high injection levels involved. In addition, extrinsic base resistance was not included in the model and base width modulation effects were omitted. As a demonstration of the effectiveness of the model for dynamic analysis in the quasi-saturation mode, the response of a common-emitter amplifier stage with sinusoidal voltage drive has been computed and compared with measured results--see Fig. 9. The type 2N3375 device, whose model parameters are given in Table 1, was used in the circuit, and the circuit conditions were such that the transistor was driven about an operating point in quasi-saturation at 100MHz. Operation was entirely in quasi-saturation (model implementation was limited to this mode). Figure 9 shows the
subroutine associated with a transport (Gummel Poon) model representation of the rest of the transistor. Alternatively the various branches of the circuit configurations of the collector region model, shown in Fig. 5, can be separately represented as circuit elements in a circuit analysis program, their currents being specified by a set of defining equations. The latter method was used in the work reported here. Regardless of the actual method of computer implementation, in order to work out the branch currents in the model at a particular time, it is necessary, for definition of the model parameters, to evaluate the width x~ of the injection region at that time. This process must in general be carried out iteratively, x~ being adjusted so that, for the instantaneous values of nodal voltages and stored charges qt, q2, there is current balance at the model nodes. At the boundary between the injection and ohmic regions of the collector, it is seen from the model of Fig. 5 and eqn (10) that x~ must be such that q2
-
(36)
Vc. + Vo
d VcB
~+(.~c dt =Ic=r~o(1-xtx,) Since C~c and ¢4 are functions of x~ (eqns 26 and 20), this relationship defines the value of x~ for specified q2, VcB and d Vcs/dt at the computation point. In a full implementation of the model, it is necessary to define operating region boundary conditions between active and quasi-saturation modes (x~ = 0) and between quasi-saturation and saturation modes (x~ = x,). In the present work, as discussed below, dynamic analysis of the transistor has been confined to the quasi-saturation region of operation. 5.2 Results with the model Typical results of representation of the d.c. characteristics of transistors in quasi-saturation by the model
5on
s~n
';"2 o.~04--
I 2
I
1
[
/L' .....
-1 2 - -
/
J,=
i
DELAY • 2 8 ns FRO . . . . . '/4 PERIOD
_
"
7
I
8
9
] 10
\
l[
-
O3-
ns
Fig. 9. Response of common emitter amplifier stage in quasi-saturation for sinusoidal base drive. Response as
computedfrom the modelfor type 2N3375devicecharacterizedby parametersgivenin Table 1. Computedvaluesof I, (mean), icm=~- icm=,,and time delay were computedas 99.3 mA, 2.7 mA, 2.8 nS respectivelyas comparedwith measured values of 96.2 mA, 2.8 mA and 3.0 nS.
Modelling of VHF and microwavepower transistors operating in quasi-saturation computed waveform of collector current which is in close agreement with measured results in both amplitude and delay. The computed delay of 2.8 nS (3.0 measured) with respect to the input voltage waveform is characteristically greater than that computed in absence of collector storage effects (2.5 nS corresponding to ~r/2 phase lag). Also shown in Fig. 9 is the "waveform" of x/xe given by the simulation. The fraction of the Nepitaxial collector region occupied by the injection region varied from 0.096 to 0.277 over the drive period. Similar experiments with a 2N1038A device, driven at 400 MHz, gave comparable results. For example, for a much smaller drive amplitude, an output delay of 1.32 nS was obtained by computation as against 1.4 nS measured (1.2 nS for Ir]2 phase lag). 6. CONCLUSIONS
A model has been developed allowing the effects of charge injection into the collector to be represented for large-signal quasi-saturation operation. This has involved the modelling of a variable-width charge storage region characterized in two-lump form. The model meets the requirements of minimum complexity for representation of the physical effects concerned, and has demonstrated its ability to simulate device behaviour under d.c., step transient and large-signal sinusoidal drive conditions, for quasi-saturation operation. Systematic methods have been presented for the determination of the parameters of the model by relatively straightforward measurement techniques. Implementation of the overall model of the transistor requires the representation of four distinct of operation, namely quasi-saturation, saturation, active, and cut-off. Only the first of these regions has been considered in this paper. In a complete model implementation, transitions between the representations for the different operating regions would need to be controlled by appropriate
SSE Vol. 25, No. 8--D
731
boundary conditions during the course of a circuit analysis computation. For quasi-saturation operation, "entry" into the model from the active region is controlled by the condition Vjc >>-O, while exit from the model (to active region) occurs when both stored charge components simultaneously become zero. At the other extreme, transition between the quasisaturation and saturation regions is, in the model, controlled by the condition xJxe = I (elimination of the ohmic section of the N-epitaxial collector). The collector region model presented here is suitable for incorporation in such an overall large-signal model capable of representing quasi-saturation operation of the transistor. P.EI~PJ~NCES I. C. T. Kirk, Electron Dev. ED-9, 164 (1%2). 2. R. J. Whittier and D. A. Tremere, IEEE Trans. Electron Dev.
ED-16, 39 (1969). 3. J. R. A. Beale and J. A. G. Slatter, Solid-State Electron. 11, 241 (1968). 4. J. A. Pals and H. C. de Graft, Philips Res. Rep. 24, 53 (1969). 5. H. C. de Graaff, Philips Res. Rep. 26, 191 (1971). 6. H. C. de Graaff, Solid-State Electron. 16, 587 (1973). 7. R. Kumar and L. P. Hunter, IEEE Trans. Electron Dev. ED-22, 51 (1975). 8. P. P. Wang and F. H. Branin, Multi-section model of an N+-P-v-N * high voltage power switching transistors. IEEE Power Electron. Spec. Con[. Record, pp. 86--89(1973). 9. F. A. Perner, Quasi-saturation region model of an NPN-N transistor. Tech. Digest of IEEE Int. Electron Dev. Meeting, pp. 418-420 (1974). 10. D. B. Treen, One-dimensional DC numerical analysis of double diffused epitaxial transistors for arbitrary injection levels. M.Eng. Thesis. Car|eton University (1971). 11. K. H. Sayeed, Behaviour of bipolar epitaxial transistors at high current levels. M.Eng. Thesis. Carleton University (1972). 12. J. R. A. Beale, E. T. Emms and R. A. Hillbourne, MicroElectronics, pp. 47-57. Taylor and Francis, London (1971). 13. A. P. Anderson and R. J. Mclntyre, High frequency high power transistor studies. RCA Victor Res. Rep. No. 6.520.2, Montreal (1965).
0038-1101[821080723-09503.0010 © 1982Pergamon Press Ltd.
MODELLING OF VHF AND MICROWAVE POWER TRANSISTORS OPERATING IN QUASI-SATURATION A. W. ALDEN CommunicationsResearch Centre, Ottawa, Canada and A. R. BOOTHROYD Department of Electronics, Carleton University, Ottawa, Canada (Received 25 March 1981;in revisedform 22 October 1981)
modellingof VHF and microwavepower transistors operating under quasi-saturation conditions is treated. Physical effects in the collector region are considered to be dominant in these devices under such circumstances, and a representation of the collectorinvolvinga variable-widthcharge-storageregion, modelled in two-lump form, is developed in order to characterize device behaviour. A systematicprocedure is presented for the evaluation of the parameters of the collector region model. The model has demonstrated its ability to describe device characteristics and performance under d.c., step transient and large signal sinusoidal drive conditions; typical results are presented. Abstract--The
1. INTRODUCTION
This paper presents a device model for the characterization of d.c. and dynamic large signal performance of VHF and microwave power transistors when operating in the quasi-saturation mode. Transistor behaviour in quasi-saturation has been considered by a number of authors[l-7]. Kirk[l] described the lowering of the reverse-bias voltage of the collector junction due to voltage drop across the ohmic region of the collector, with the junction becoming forward biased at the higher current levels. Beale and Slatter[3], and Pals and de Graaff[4], amongst others, considered injection of minority carriers into the collector under these conditions and developed physical models for d.c. operation. More recently, it has been recognized [8, 9] that a singlelump model of the collector region is unable to represent adequately such injected minority carrier charge under large-signal transient conditions; the models developed, however, do not specifically model the storage of charge in a finite region of the collector. In the present work, a lumped model of the collector region is developed which corresponds with the physical conditions that exist in quasi-saturation, and which characterizes behaviour of the device for d.c. and dynamic operation. Methods are given for determining the defining parameters of this model by measurement. The devices under study are illustrated in Fig. 1 devices of overlay type with interdigitated and mesh emitter structures. These device structures are designed to minimize edge effects around the emitter, and can reasonably be treated as a parallel combination of many, essentially one-dimensional, intrinsic transistors for purposes of analysis. The definition of an "intrinsic" device, as defined here, is indicated in Fig. 1. Specific transistor types considered, for which results are reported, are 2N1038A and 2N3375, of 1.1 GHz and 400 MHz .fT respectively.
For the types of transistor under consideration, detailed computer simulation[6, 7, 10, 11] has shown that under conditions of quasi-saturation a narrow space charge region exists in the vicinity of the base-collector metallurgical junction, corresponding to a forward-biased condition[l, 3, 4], and that the N - epitaxial collector may be divided into two distinct regions, as indicated in Fig. 2: (a) an approximately neutral injection region adjacent to the collector-base junction, over most of which conditions of high-level injection exist; Co) an ohmic region, contiguous with (a) and extending to the N + collector region, operating under conditions of negligible conductivity modulation. The excess minority carrier charge stored in (a) is generally much larger than that stored in the base, and is responsible for major dynamic effects. This charge is constrained within the injection region (a) by the electric field associated with the potential drop across the ohmic region (b) due to majority carrier flow, which is responsible for the resultant forward bias applied across the collector-base junction. The potential drop across the injection region, caused by the electric field distribution associated with high level injection, is very small. For given collector-base voltage, as the collector current is increased, corresponding to increased emitter-base forward bias, the injection region (a) extends farther towards the N + collector contact, and the ohmic region (b) shrinks in width. The two extremes of the quasisaturation regime are where the N - collector region is entirely "injection" on the one hand, and entirely "ohmic" on the other. The modelling problem is thus to represent a neutral charge storage region of variable width, depending upon bias conditions, coupled to an adjacent ohmic region. In the following sections, the collector region model is developed from d.c. relationships in quasi-saturation, 723
A. W. ALDENand A. R. BOOTHROYD
724 [.
AREA AC
~ABEA ~ A E = BASE CONTACT DIFFUSION I EMITTER
P
with the current transported through the region. On the basis of one-dimensional current flow, throughout the collector region
] I I
] I
I
I'
J = Jr, + Jp = constant
SE
I
I
I [ I
I N EPITAXiAL COLLECTOR , I ' I II
I"
EXTRINSIC DEVICE
I
since both 3", and Jp must be continuous and Jp is negligible in the ohmic region. The injection region, essentially neutral, contains equal excess charges of electrons and holes resulting from the forward bias of the collector-base junction, but flow through this region is by electron transport only.
LI
Fig. 1. Cross section of interdigitated power transistor structure.
2.1 Injection region
In the injection region, 0 ~
COLLECTOR/BASE JUNCTION REGION
"t
°Hct
REGION R
REGION
IB
(2)
Xi
&
+ ~
---'wv--
C
z ==
Xe
_ kT l dp q p dx
kT d(ln p) q dx
(3)
Upon substitution into the expression for Jr,, with the high level and neutrality conditions (p=n>>NepO assumed, the electron current density is given by the familiar high level expression
.x
J~ = q( 2/9, ~dn = - IJAE
Fig. 2. Definition of injection and ohmic regions of epitaxial collector. with a two-lump representation of charge storage in the injection region. Methods of determining the parameters of this model from appropriate measurements are presented, and its d.c. and large signal dynamic performance are illustrated.
(4)
where AE is the emitter area. The associated distribution of excess hole and electron density is triangular, of slope (JJ2D,,q), as illustrated in Fig. 3. The expression of eqn (4), and the triangular distributions of excess stored charge, apply over most of the injection region, losing validity close to x = x~. From eqn (4), the corresponding hole density p(0) at the junction boundary x = 0 is
2. d.c. MODEL OF COLLECTOR REGION IN QUASI-SATURATION
The device region under consideration is here referred to as the "intrinsic collector" as illustrated in Fig. 1. This region is assumed to be part of a one-dimensional structure, as shown in Fig. 2, of cross sectional area AE equal to that of the emitter and extending through emitter, base and collector. The N - collector region, of uniform donor density Nop~ and total width x,, is regarded as divided into an injection region adjacent to the junction, and an ohmic region. In the injection region, high level injection is assumed, so that p,n >>N~pi. In the ohmic region, negligible conductivity modulation is assumed with n = Ncp~. The inconsistency of these assumptions at the boundary x = x~ between the two regions implies a gradual transition from high to low injection conditions in the vicinity of x~; i.e. between p>>N~p~ in the injection region and p "~Nepi in the ohmic region. A reasonable definition of the boundary x = xl, adopted here, is thus
p(x~) =
Noel.
p(O) = Lx,/2D, qA~.
(5)
The hole density of eqn (5) has to be in agreement with the value given by the injection law of the collector-base junction. On the basis of low-level injection conditions in the base, ni2 p(0) = ~ exp (q Vjc/kT)
(6)
where Vjc is the voltage applied to the base side with
=
pl,n~
Jn
-Dnq
(1) I
The starting point in the analysis and d.c. modelling of the collector region is the assumption that recombination current is suflicently small to be neglected in comparison
x,
Fig. 3. Carrier density distributions in the collector injection region.
Modelling of VHF and microwave power transistors operating in quasi-saturation respect to the collector side of the junction. Simulation studies[10, 11] have shown that over a wide range of quasi-saturation conditions, the collector injection region can be in high injection while the base region operates at low level, so that eqn (6) is applicable. Otherwise, this relationship needs to be modified to allow for high level effects in the base. Owing to the presence of the (negative) electric field caused by carrier injection, a potential V(xD is developed at x~ with respect to the junction region boundary x = 0, where
f ; ' ~ kT 1 dp l
V(xD
i
=
1--q-p TxxJdx
kT q In [[ pp(O) -~)j
(7)
Using the junction law, eqn (6) together with the definition of eqn (1) for the location of the boundary x = x~ between the injection and ohmic regions, we obtain with Beale et al. [12], V(x,) = V , c - Vo
(8)
where
v°=k-~-Tq ln(N-m'}2"\n~ /
(2)
With the equilibrium condition as reference, the potential of the boundary at x = x~ with respect to the base contact, neglecting base resistance, is thus - Vo. 2.2 Ohmic region An externally applied bias voltage Vca results in a voltage VcB+ Vo across the ohmic region. Thus,t assuming negligible conductivity modulation, V~,,+ I,'o V ' ~ + Vo Ic - x, - x-~ " qlz"NcpiAE = r~o(1 - x~x,)
is required that
2 D n'(O). A~ q " x~
VcB + Vo r~o(1-x.Jx,)
(11)
where n'(0) is the excess electron density at the collector junction boundary. 3. DYNAMIC CHARGE STORAGE MODEL OF COLLECTOR
Under conditions of quasi-saturation operation, charge storage in the injection region of the collector exerts a major influence on the dynamic behaviour of the device. The charge stored in this region can be considerably greater than that stored in the base. Adequate modelling of the dynamic charging effects associated with this region is therefore important, and should characterize with sufficient accuracy the distribution charge flow in a storage region of time-dependent width, with a minimum of model complexity. As in the comparable situation of charge storage in a P N junction diode, a two-lump charge storage representation of the region is chosen as being the simplest form of model which can meet these requirements. Conditions in the two charge storage lumps are defined by the boundary conditions at x = 0 and x = x, together with the requirement of continuity of current flow at their common boundary. 3.1 The basic two-lump model As indicated in Fig. 4, the injection region is divided into two storage lumps, of widths x~, x2 containing excess majority carrier charges ( - ql), ( - qz), where the ratio
K = xl/x2
(12)
can be chosen arbitrarily. These stored charges are related to the average excess majority carrier densities nl' -- ql/qAExl n2 - q2/qAex2J
(10)
I
~
where rco is the resistance of the full ohmic region when x~ = 0. For given conditions of operation VcB, Ic in quasi-saturation, eqn (10) defines the position of the boundary x = xi. The current expression eqn (4) in the injection region then specifies the hole injection level p(0) at the collector junction boundary, as in eqn (5), and hence the forward bias voltage of the collector-base junction by eqn (6). The same value of lc needs to be specified by the base region model, with an appropriate choice of applied VsE or of base current. The modelling problem is that of representing the current flow and associated charge storage properties of a collector region of variable width x, where xi is such that eqns (4) and (10), for the collector current flowing through the injection and ohmic regions respectively, are simultaneously satisfied. Thus, for the d.c. steady state, it
725
1 I
Nep i
(13)
•
I I
I
X1
Xi
× Xe
(a)
n
i
I Xe
X
XI÷X 2 (b)
tBase resistance rw is ignored in this discussion; strictly, with base resistance included, Vc8 should be replaced by VcB,.
2
Fig. 4. Definition of 2-lump charge storage model of the collector injection region.
726
A.W. ALDENand A. R. BOOTHROYD
respectively for the two lumps. For given charges and value of x. the current I, flowing from lump 2 to lump 1 (corresponding to electron flow in the opposite direction), and the current/2 flowing from the ohmic collector region into lump 2, may be expressed by difference equations in terms of n'~ and n~', these being regarded as equivalent electron densities at the centre point of the respective lumps. Thus, for high-level transport of majority carriers, neglecting recombination effects, L
-
q A E ( n l - n~). 2D. (x, +x2)/2
(14)
qAnn~. 2D, x2/2
(15)
and
12=
In terms of the corresponding stored charges, = (q,-Kq2)/¢3
(16)
= q2/¢4
(17)
with inclusion of the charging currents q~, q2 of the storage lumps and, as a second order effect, recombination currents ql/rl, qJ'r2 (where ¢~, ~'2">r3, ~'4). The current C~(~cB, which represents charging via the ohmic region boundary, is treated in Section 3.2. 3.2 Dynamic boundary conditions--region width modulation Under dynamic conditions, the currents associated with the stored charges q~, q2 are as discussed above and represented in Fig. 5. Thus, the current flowing from Jump 1 of the collector model into the base is (I, + qt/~ + q~), and must be consistent with the current value given by the base region model for the applied junction voltages. However, in addition to the currents so far considered, which correspond to a specified region width x, it is necessary to represent the charging current flowing at the ohmic region boundary of the injection region (x = x~) due to variation of this width. For the toal stored charge q~( = q~ + q2) in the injection region, we may write dqc
where the "transit time" parameters r3, ~'4 are r3 = r3o(XJXe) 2
(18)
x, 2 K r3o = 4/), " I + K
(19)
r , = r,o(XJX+) 2
(20)
with
and
I,, = ~
z+o K(1 + K)'
(21)
Figure 4 illustrates these relationships for the d.c. steady state, in absence of recombination of excess charge; in this case, evidently I, = I2. Figure 5 shows the model representation of the collector injection region under general dynamic conditions,
[ .......
~q" I
vsc........
~ _ (ql_Kq2)/.t,3
Ic I ]
q~
ql/rl q2
HOLE FLOW INJECTIONREGION JUNCTION REGION
d Vcs = c d Vc~ dt ~,c dt
C,c - - q" n'(O)Ae . dxi 2 d Vcn"
C
rc [ BASE
(22)
(23)
This charging current is included in the model of Fig. 5. The capacitance C,+c, evaluated from the static triangular charge distribution in the injection region (with recombination effects neglected) is
C.C dVcB dt ELECTRON FLO~W
aq¢ ax~ t q" ax~ at
where the second term is the charging current entering the injection region through the boundary at x = x+ due to change in region width alone (constant boundary conditions at x = 0). In the model, this charging current flows into lump 2, and is evaluated on the basis of quasi-static conditions. Accordingly, we define a charging current I,+c in terms of the change of charge dqc resulting under static conditions with the collector junction voltage constant and the injection region width changed by dx+ due to a change d VcB in collector voltage, in the same manner as in the modelling of base width modulation, and write
with T3O
aqc
"~'-=-"~'lxi
q /r
I I I I I I OHMIC 19
REGION
Fig. 5. Lumped model of the collector for quasi-saturation operation.
(24)
Modelling of VHF and microwavepower transistors operating in quasi-saturation From eqn (11), dx~ d Vc~
-x~ 2
2qAED, n'(O)r~o
(25)
giving
x~2 =C c~c = 4D.rco
( x~]z ,,CO~xS/
(26)
727
specifications, Vo, and r~o are known; otherwise, separate determination of these parameters is necessary. It is shown below that, with K specified, the model parameters can be deduced almost entirely from d.c. characteristics of the device, supplemented by an observation of transient behaviour when the device is switched between different quasi-saturation states. Typical results are presented for type 2NI038A and 2N3375devices. 4.1 Parameter determination from d.c, measurements
where
1 + K. r3_..~o C"~°=
K
(27)
rco
3.3 Complete device model A complete device model incorporating the above modelling of the "intrinsic collector" can be developed along conventional lines. Neglecting fringing effects, the remainder of the collector (outside the intrinsic region) may be regarded as being purely ohmic with the collector junction reverse-biased. This region has collector series resistance t__
A~
rc - r~o • A c - A ~ '
(28)
By interpretation of the d.c. characteristics of the transistor in terms of the relationships presented above, all the parameters needed for d.c. characterization of the collector model can be obtained. Three sources of data are utilized relating to the following features of the device characteristics in quasi-saturation: saturation] quasi-saturation boundary ( I c - V c e locus), active/quasisaturation boundary, common-emitter current gain in quasi-saturation. The saturation/quasi-saturation boundary locus across the common-emitter collector characteristics can be estimated, as indicated in Fig. 6 (e.g. points a, b, c). Along this locus, since rc = 0 and the injection region occupies the entire collector epitaxial region, Vo = VcE + VEB + IBrbb' = Vca'.
where A c is the total collector junction area and area ( A c - AE) carries only collector junction charging current. The base and emitter regions of the transistor are suitably modelled for example by a Gummel-Poon transport model in conjunction with an extrinsic base resistance. 4. DETERMINATION OF THE PARAMETERS OF THE MODEL
For the representation of quasi-saturation operation of the transistor, an intrinsic collector region model has been presented which is specified by the following parameters: Parameter
Definition
Vo(or Nepi)
eqn (9) eqn (10) eqn (12) eqns (19) and (21) Fig. 5 eqns (26) and 27)
rco(Orx,) K ¢3o,~4o r~, ~'2 C~co
These parameters need to be determined for a given device, in addition to those of the rest of the complete device model. We shall here be concerned with procedures for determining the above parameters of the intrinsic collector model, the remaining model parameters being assumed evaluated by familiar methods of measurement. Of the parameters listed, the "lumping parameter" K is arbitrary. The choice K = 1 is made here, corresponding to the injection region of the collector divided into two lumps of equal width. If the epitaxial region parameters Nepi, xe are known from device design
(29)
Consistent values for Vo have been obtained for samples of both types of device studied (2NI038A and 2N3375) over a range of points along the boundary locus. For 2N3375 devices, good agreement has also been obtained with Vo values calculated from eqn (9) for a known N,p~ value of 1.6× 10~cm-3: e. g. at 25°C, Vo=-0.60 from eqn (9) as compared with an average value of - 0.59 from the d.c. characteristics, with a spread of values of -+5% in the latter for the range of locus points taken. Once Vo is found, the collector resistance for any bias point in quasi-saturation follows from eqn (10), viz. rc
v¢~, + Vo - = rco(l - x~x,). Ic
(30)
The boundary locus between the active and quasisaturation regions (shown as 1, 2, 3 on the collector characteristics) of Fig. 6 can be used in order to determine rco from eqn (30) since along this locus the collector epitaxial region is entirely "ohmic" and xi =0. Results for rco consistent to within 10% accuracy have been obtained for the 2N1038A and 2N3375 devices using points along this locus, giving average values of 5.3 and 8.51"1, respectively. The result for the 2N3375 compares reasonably with 9fl determined by Anderson et al.[13] using an admittance bridge method, and 11.6[1 calculated from available design data for this type of device (emitter area = (12.7 g,m)2x 160, x¢ = 10•m, Nepi = i.6 x 10'5cm-3). With rco known, the ratio xdx, follows from eqn (30) for any bias point in quasi-saturation. At this stage in the parameter evaluation process, we are left with the characteristic times ~'3o, ~'~o, r~, ¢, and
A. W. ALDENand A. R. BOOTXROYD
728
160--
giving
I
140~
Ic = Ic(1 + llflF) Is ~o. Is - IciEr
/
2N3375
120--
100--
_o
where /3F is regarded as known (e.g. measured at the same lc value for Vcs, = 0). This current ratio can also be expressed in terms of the model parameterg. Referring to Fig. 5, we have
2
SATURATION
IBm2 mA
80--
(33)
// ~/~ATU"ATION/-'/4~
i-QUASI-/ /
Ic = q2]7-4
60-and with r, = 7-2,
40-20-0
In coil= ql + q2
TI
f// / I
I
I
I
I
I
I
02
04
0.6
0 8
1.0
1 2
1.4
Now, for d.c. current continuity within the model
VCE VOLTS
Fig. 6. Collector characteristics of 2N3375 transistor, defining operating regions (measured). the capacitance C,~o yet to be found. It should be noted that for the modelling of d.c. relationships, only the ratios of the "time parameters" are required. Thus, making the reasonable assumption that the recombination lifetimes r~, r2 for the two charge storage lumps are equal, only the values of the ratios r4o/7-3o, rl]7-3o need to be known fro purposes of d.c. characterization. However, from the defining relationships of the twolump collector charge storage model, with K chosen as unity, we have (see eqn 21).
so that q_2+ K(1 + K) + K + 2
q2
7"1
and Ic rt K ( K + 1) Is co----~i- 7-3 {1 + K ( K + 2) + r317-,}"
Using the condition rl "> 7-3,with K = 1, we obtain
/C
IB~o. 1
1
Z4oh'ao K ( K + 1 ) - 2
(31)
so that only r,/7-3o remains to be evaluated. The common emitter current gain properties in quasisaturation provide a source of data from which this remaining d.c. parameter can be deduced. In quasisaturation, the current can be written IB = IB ~
+ Io col,
(32)
where the second term originates in the collector injection region. The current gain is reduced by only a modest factor compared with active region operation. Thus, since for the type of device under consideration Neo~ is small compared with the level of base doping, it is reasonable to expect the base current component originating from reverse injection into the base to be of significantly smaller magnitude than either IB b,~ or IB c,,,. Hence, with reference to Fig. 5, 1. h..o = (1c + 1~ co.)//~F and I . ~o,I = I . - ( I c + I~
co.)l~
(34)
7-1
2r{
(35)
The parameter ratio 'rl/'T 3 c a n therefore be found for any point in quasi-saturation by utilizing eqns (33) and (35). Having previously determined rco, xJx, is also known for this bias point from eqn (24); the ratio 7-~/Z3ois known from eqn (18). The process of d.c. characterization is complete at this stage. Figure 7 shows typical agreement obtained between measured and computed collector characteristics for a 2N3375 device, using the model relationships and the parameter values listed in Table 1. The parameter values obtained for the 2N1038A device are given in Table 2. 4.2 Parameter determination [rom collector transient response The parameter determination process is completed by the evaluation of the "transit time" parameter 7-3o,which characterizes current transport, through the carrier injection region of the collector. The width modulation capacitance parameter C, co, related to 7-30by eqn (27) is then also known. Thus, the dynamic behaviour of the collector region in the quasi-saturation mode is dependent upon this single parameter as far as time scale is concerned, all other relationships having been established by d.c. measurements.
Modellingof VHF and microwavepower transistors operatingin quasi-saturation
160-
729
MEASURED 136-
2N 3375 J
""
i
134IcmA (COMPUTED) 132--
140X i/Xe = 0
120--
~#~,,-,~
130--
//r3°=6° ns j r / r 3 0 = 6 ns .
BOUNDARY ~ , " "
100-
.
.
.
.
.
.
.
.
.
.
IC-2 mA 128(MEASURED) - 126 --
. - " " "-
E 80-
124
I o
I 1 t ,,sec
I 2
60-
///f-
40-
Fig. 8. Collector current response to a collector voltage step while in quasi-saturation.Comparisonof response of the model with measurementfor type 2N3375device.
........
l/
EABU EO
20-
['
.........
?gg2E RA TEnB TABLE I1
I 0.2
0
I 0.4
I 0.6
I 0.8
I 1.0
I 1.2
I 1.4
VCE VOLTS
Fig. 7. Comparison of collector characteristics computed from the model with those measured, for type 2N3375 device sample.
Table 1. Collectormodelparameter values: 2N3375 P a r e
Value
v 0
-0.6 volt
rco
8.5~
K
1.0
x40/x30
0.5
Xl/X30
78.6
x 2/x30
78.6
•t
30
4.3aS
Table 2. Collectormodelparameter values: 2N1038A parmmter
Value
V0
-0.6 ~olt
•co
5.3fl
g
1.0
x4~x30
0.5
Xl/X30
340
x~x30
340
x30
86pc
The value of r3o may be deduced from the transient response of collector current when the transistor is "switched" from one quasi-saturation state to another by applying a step voltage drive to the collector, with the transistor in common-emitter configuration and base current maintained constant. When this is done, the transistor undergoes a transition between states corresponding to different values of injection region width
x, or of charge storage in the collector. The method used is to match the collector current response given by a computer implementation of the transistor model to the response observed in the test on the device, by adjusting the one remaining "free" parameter ~'3oof the model for the best agreement. Figure 8 illustrates the typical comparison of measured and computed collector current response waveforms, with several values of r3o, for a device sample of type 2N3375. With 18 = 3mA, a step change of Vc~ from 0.80 to 0.85 V is applied, taking the transistor to a state less deeply in quasi-saturation. Because of the relatively small changes in Vcr and Ic, the response waveforms of Fig. 8 are aligned to eliminate the initial discrepancy in lc between the model and the actual device. The best choice of r3o was judged to be 4.3 nS, which is in good agreement with the value of 4.17 nS calculated from eqn (19) with the design value of 10 #m for x,. The values of the parameters rso, ~',,o, r,, r2, and C, co presented in Table 1 for the type 2N3375 device all followed from the determination of r3o by the transient test method, taken in conjunction with the parameters previously obtained from d.c. measurements (as did for those for the 2N1038A device in Table 2). The transient test method can also be used to determine the value of collector resistance re. Since the collector injection region width x~ cannot change instantaneously on application of a step change A Vcs, there is an initial step change A VcB[rc of collector current as shown in Fig. 8. Values of r~ obtained by this transient test method have been found to agree with those from d.c. measurements as described in Section 4.1. Use of the transient test method is advantageous where the boundary between the saturation and quasi-saturation regions is ill-defined on the d.c. characteristics. Under these circumstances, determination of Vo from this boundary (see eqn 29) is unreliable and an alternative procedure is to use the d.c. relationship of eqn (30), with rc determined from the transient test, to obtain the value of Vo. 5. MODEL IMPLEMENTATION AND RESULTS
5.1 Implementation For purposes of d.c. or large-signal dynamic analysis, the collector region model defined by the relationships presented above can be best implemented as a computer
A. W. ALDENand A. R. BOOTHROYD
730
were presented in Section 4 in connection with model parameter determination. Figure 7 shows examples of the degree of matching obtained between measured characteristics and those given by the model. The ability of the model to represent dynamic behaviour of the collector region is indicated by the results shown in Fig. 8 for collector current response to a step change in VcB. The close agreement with the measured response gives confirmation of the adequacy of representation by the model of the charging conditions at the boundaries of the collector injection region and the consequent change of its width with time; the initial choice of a two-lump representation of this region, and the application of charge control relationships appear to be well justified. In the implementation of the model used to obtain the above results (and those discussed later in this section), certain simplifying approximations were made, justified by conditions referred to in earlier sections, namely: (i) forward injection dominant in the base region (reverse transport current neglected); (ii) depletion layer charging currents neglected in comparison with injection charging currents owing to high injection levels involved. In addition, extrinsic base resistance was not included in the model and base width modulation effects were omitted. As a demonstration of the effectiveness of the model for dynamic analysis in the quasi-saturation mode, the response of a common-emitter amplifier stage with sinusoidal voltage drive has been computed and compared with measured results--see Fig. 9. The type 2N3375 device, whose model parameters are given in Table 1, was used in the circuit, and the circuit conditions were such that the transistor was driven about an operating point in quasi-saturation at 100MHz. Operation was entirely in quasi-saturation (model implementation was limited to this mode). Figure 9 shows the
subroutine associated with a transport (Gummel Poon) model representation of the rest of the transistor. Alternatively the various branches of the circuit configurations of the collector region model, shown in Fig. 5, can be separately represented as circuit elements in a circuit analysis program, their currents being specified by a set of defining equations. The latter method was used in the work reported here. Regardless of the actual method of computer implementation, in order to work out the branch currents in the model at a particular time, it is necessary, for definition of the model parameters, to evaluate the width x~ of the injection region at that time. This process must in general be carried out iteratively, x~ being adjusted so that, for the instantaneous values of nodal voltages and stored charges qt, q2, there is current balance at the model nodes. At the boundary between the injection and ohmic regions of the collector, it is seen from the model of Fig. 5 and eqn (10) that x~ must be such that q2
-
(36)
Vc. + Vo
d VcB
~+(.~c dt =Ic=r~o(1-xtx,) Since C~c and ¢4 are functions of x~ (eqns 26 and 20), this relationship defines the value of x~ for specified q2, VcB and d Vcs/dt at the computation point. In a full implementation of the model, it is necessary to define operating region boundary conditions between active and quasi-saturation modes (x~ = 0) and between quasi-saturation and saturation modes (x~ = x,). In the present work, as discussed below, dynamic analysis of the transistor has been confined to the quasi-saturation region of operation. 5.2 Results with the model Typical results of representation of the d.c. characteristics of transistors in quasi-saturation by the model
5on
s~n
';"2 o.~04--
I 2
I
1
[
/L' .....
-1 2 - -
/
J,=
i
DELAY • 2 8 ns FRO . . . . . '/4 PERIOD
_
"
7
I
8
9
] 10
\
l[
-
O3-
ns
Fig. 9. Response of common emitter amplifier stage in quasi-saturation for sinusoidal base drive. Response as
computedfrom the modelfor type 2N3375devicecharacterizedby parametersgivenin Table 1. Computedvaluesof I, (mean), icm=~- icm=,,and time delay were computedas 99.3 mA, 2.7 mA, 2.8 nS respectivelyas comparedwith measured values of 96.2 mA, 2.8 mA and 3.0 nS.
Modelling of VHF and microwavepower transistors operating in quasi-saturation computed waveform of collector current which is in close agreement with measured results in both amplitude and delay. The computed delay of 2.8 nS (3.0 measured) with respect to the input voltage waveform is characteristically greater than that computed in absence of collector storage effects (2.5 nS corresponding to ~r/2 phase lag). Also shown in Fig. 9 is the "waveform" of x/xe given by the simulation. The fraction of the Nepitaxial collector region occupied by the injection region varied from 0.096 to 0.277 over the drive period. Similar experiments with a 2N1038A device, driven at 400 MHz, gave comparable results. For example, for a much smaller drive amplitude, an output delay of 1.32 nS was obtained by computation as against 1.4 nS measured (1.2 nS for Ir]2 phase lag). 6. CONCLUSIONS
A model has been developed allowing the effects of charge injection into the collector to be represented for large-signal quasi-saturation operation. This has involved the modelling of a variable-width charge storage region characterized in two-lump form. The model meets the requirements of minimum complexity for representation of the physical effects concerned, and has demonstrated its ability to simulate device behaviour under d.c., step transient and large-signal sinusoidal drive conditions, for quasi-saturation operation. Systematic methods have been presented for the determination of the parameters of the model by relatively straightforward measurement techniques. Implementation of the overall model of the transistor requires the representation of four distinct of operation, namely quasi-saturation, saturation, active, and cut-off. Only the first of these regions has been considered in this paper. In a complete model implementation, transitions between the representations for the different operating regions would need to be controlled by appropriate
SSE Vol. 25, No. 8--D
731
boundary conditions during the course of a circuit analysis computation. For quasi-saturation operation, "entry" into the model from the active region is controlled by the condition Vjc >>-O, while exit from the model (to active region) occurs when both stored charge components simultaneously become zero. At the other extreme, transition between the quasisaturation and saturation regions is, in the model, controlled by the condition xJxe = I (elimination of the ohmic section of the N-epitaxial collector). The collector region model presented here is suitable for incorporation in such an overall large-signal model capable of representing quasi-saturation operation of the transistor. P.EI~PJ~NCES I. C. T. Kirk, Electron Dev. ED-9, 164 (1%2). 2. R. J. Whittier and D. A. Tremere, IEEE Trans. Electron Dev.
ED-16, 39 (1969). 3. J. R. A. Beale and J. A. G. Slatter, Solid-State Electron. 11, 241 (1968). 4. J. A. Pals and H. C. de Graft, Philips Res. Rep. 24, 53 (1969). 5. H. C. de Graaff, Philips Res. Rep. 26, 191 (1971). 6. H. C. de Graaff, Solid-State Electron. 16, 587 (1973). 7. R. Kumar and L. P. Hunter, IEEE Trans. Electron Dev. ED-22, 51 (1975). 8. P. P. Wang and F. H. Branin, Multi-section model of an N+-P-v-N * high voltage power switching transistors. IEEE Power Electron. Spec. Con[. Record, pp. 86--89(1973). 9. F. A. Perner, Quasi-saturation region model of an NPN-N transistor. Tech. Digest of IEEE Int. Electron Dev. Meeting, pp. 418-420 (1974). 10. D. B. Treen, One-dimensional DC numerical analysis of double diffused epitaxial transistors for arbitrary injection levels. M.Eng. Thesis. Car|eton University (1971). 11. K. H. Sayeed, Behaviour of bipolar epitaxial transistors at high current levels. M.Eng. Thesis. Carleton University (1972). 12. J. R. A. Beale, E. T. Emms and R. A. Hillbourne, MicroElectronics, pp. 47-57. Taylor and Francis, London (1971). 13. A. P. Anderson and R. J. Mclntyre, High frequency high power transistor studies. RCA Victor Res. Rep. No. 6.520.2, Montreal (1965).