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Large-signal analysis of the silicon pnp-BARITT diode
Solid-State
Electronics. 1976. Vol. 19. pp. 625-631.
Pergamon Pres.
Prmted in Great Bntaio
LARGE-SIGNAL ANALYSIS OF THE SILICON pnp -BARITT DIODE M. KARASEKt Department of Electronic and Electrical Engineering, University of Birmingham,BirminghamB152TT, England (Received 31 October 1975;in
revisedfonu
23 January
1976)
Abstract-A numerical large-signal analysis of the silicon pnp BARITT diode is presented. This extends previous works to practical device structures. The diode admittance, power generation density, efficiency and quality factor are studied as functions of oscillation amplitude, frequency and bias current density. The results are found to be in substantial agreement with large-signal experimental measurements. A typical pup structure optimised for operation at 10GHz has an active region doping density of 2 x lO”/cm’, a width of 5.5 pm and a punch-through voltage of 40 V. Negative resistance occurs over the frequency range from about 8 GHz to about 14GHz. The device Q at 10GHz varies from a small-signal value of about -10 to a large-signal value of about -100. The conversion efficiency at IOGHz is about 5% at a bias current density of 25 A/cm* and falls to about 2% at 200A/cm*. Maximum r.f. power outnut occurs at an r.f. amolitude of about 15V and rises with bias current density to a maximum value of about 176W/cm2at 200A/cm’. L 1. INTRODUCTION
1%8 Wright[l] described a new negative resistance microwave device based on the principle of barrier controlled injection and transit time delay-the BARITT diode. His simple analysis suggested that the device should operate at moderate power and low noise levels. In the same year, independently, Ruegg presented a paper [2] on the simplified large-signal theory of a similar punchthrough structure giving considerably optimistic prospects-an estimated efficiency of the order of 20% and power output of 10-100 W at 10GHz. These theoretical works were confirmed experimentally in 1970 when Sultan and Wright [3] achieved negative resistance in npn silicon structures, and subsequently oscillations in pnp structures[4] and in 1971 when Coleman and Sze[S] reported oscillations in metal-semiconductor-metal structures. Several experimental papers have since been presented [6,7,8] comparing the properties of different BARIn diode structures and pointing out the reliable and low-noise operation of the device at moderate power levels. Following the achievement of microwave oscillations, extensive theoretical analyses and numerical calculations have been reported discussing the small-signal behaviour of various BARITT structures [9,10,11] and providing useful insight into the physical and electronic mechanisms of their operation. However, for utilization of the BARITT diode as a free-running oscillator, modulated source or self-oscillating mixer the knowledge of the large-signal properties is of vital importance. Unfortunately only a few contributions are available on the large-signal analysis of BARITT diodes[2, 12-141. The main reason seems to be the difficulty of solving the nonlinear system of partial differential equations governing carrier injection and transport. Even if a onedimensional model is used, it is impossible to obtain a In
closed analytical solution without severe simplifications such as space-charge-limited injection, neglect of carrier diffusion and the assumption of saturated carrier velocity throughout the whole active region. Such an analysis has been made by Wright[lS] and can give useful and informative first-order results. The aim of this paper is to present a numerical technique which has been used successfully for the large-signal analysis of the BARITT diode with two pn junctions and which takes into account all the necessary practical physical conditions such as a realistic doping profile, a suitable carrier velocity-electric field relation, and both drift and diffusion components of particle current. The computer programme which has been developed has been applied to a typical X-band pnp silicon BARITT diode and large-signal values of admittance, power generation density and efficiency are shown to be dependent on oscillation amplitude, frequency and bias current density. The time and space dependence of the electric field intensity and positive hole density in the case of largesignal oscillations have also been studied. Despite the relative complexity of the basic equations the programme is not extremely computer time consuming and may be used for parameter optimization of pnp, pnup and similar BARITT structures with two pn junctions. 2. MATHEMATICAL MODEL
Although the mathematical descriptions of the processes inside different BARITT diode structures are very similar we restrict ourselves in the following analysis to a pnp structure. This is the most basic design and is the most practical for device fabrication. In the active state the central transit region of this structure is completely depleted and the transport of injected carriers is described by Poisson’s equation, the continuity equation and Maxwell’s equation of total current together with the equation of particle current density. These equations are as follows
ton leavefrom Institute of RadioEngineeringand Electronics, CzechoslovakAcademy of Sciences, 180 88 Praha, Czechoslovakia.
dE(x,t) dX
625
e
=T[p(x,t)-NA(x)+No(x)l
(1)
M. KARASEK
626
G,(x) g [PdX)P(X)~
t)v
A&
[2-pK(Ax)*]Ai,+A$K+I=A
I-
where E X f! E
P NA, ND t J, JT l-h D,
BpwtCpK+Dp,c+I=O
electric field intensity, space coordinate, electronic charge, permittivity of semiconductor material, positive hole density, density of acceptor and donor impurities, time, hole current density, total current density hole mobility, hole diffusion coefficient.
-
(pK
+ pK
,,)/2
exp
K = 2,3,4. .
ax
J,(x,t)=p,(x,f)
pkt)E(x,t)-jy-
aP(x, t)
I
(8)
The diffusion constant in eqn (4) was substituted in terms of the carrier mobility by means of the well known Einstein relation.? The Caughey-Thomas formulae [ 171 have been used for the carrier velocity-electric field relations. To complete the specification of the mathematical model a set of initial and boundary conditions must be defined. The electric field spatial distribution corresponding to a given d.c. bias is taken as the initial condition for the subsequent large-signal analysis. When computing the d.c. field distribution the original set of partial differential equations is converted into two ordinary differential equations as follows dE(x) dx
-=p(x)+
+This is justified because importantin thoseregionsandat small.
(h
-
4K+d
1499.
(13)
The above sets are solved iteratively together with boundary conditions which, under the assumption of infinite recombination contact regions, are in the form E(x,t)=O p(x,t)=N(x)
__ aJ,(~~t)
at
(12)
D =(/.LcLK +pKtl)/2exp(rlrK-l-IlrK)
=tx, t) -=p(x,t)+N(x) ax t)_
(11)
where AeK is a correction to the electrostatic potential. The coefficients A, B, C, and D are defined as follows:
In order to simplify the mathematical manipulations and to avoid possible numerical overflows and underflows during execution of the programme it is convenient to normalize the above set of equations into a dimensionless form. The normalization factors used are the same as those proposed by DeMari [ 161.When keeping the same symbols the normalized equations are
a~0 A_
(10)
where (p,, is the quasi-Fermi level for holes. These nonlinear equations are linearized by the NewtonRaphson method and the usual finite difference technique is applied leading to two sets of linear algebraic equations
40, t) = wpk t)p(x, t)Ek f) -eD,(x,
I= 0
x =O,L.
(14)
In order to demonstrate the calculation of initial conditions several results of the d.c. analysis are presented. A typical pnp device structure was chosen designed for operation at X-band. This is shown in Fig. 1. The transit region was 5.5 pm wide and was uniformly doped with a donor density of ND = 2 x lOI5cm-‘. The calculated d.c. current-voltage characteristic of the diode is shown in Fig. 2. When the voltage is below the punch-through value ( V,, = 39 V) there is a sufficiently high potential barrier in front of the source pn junction that only a small amount of holes is injected into the transit region and almost no current flows. The spatial distributions of hole density and electrostatic potential in the vicinity of the source pn junction are displayed in Fig. 3. When the bias voltage
N(x)
the effects of diffusion are only those timeswhen theelectricfieldis
DISTANCE
iflrnl
Fig. I. Doping concentration profile used for the silicon pnp BARITT diode.
Large-signal analysis of the silicon pnp-BARITT diode
40;
i 0
__I._ 50 CURRENT
100DENSITY
--x7 (A/cm’)
200
Fig. 2. Calculated dc. current-voltage characteristic.
POTENTIAL
Y 0 I
1
L.00,
\,
IO8
Fig. 3. Distribution of d.c. hole density and electrostatic potential in the vicinity of source pn junction.
exceeds 39V, the potential barrier of the source pn junction begins to diminish, the amount of injected holes increases and the diode current sharply rises. Figure 4 shows the static spatial distributions of the electric field and hole density throughout the structure corresponding to a d.c. current density J, = 100A/cm2. In order to study the large-signal performance of the BARITT diode at a given frequency but for various bias conditions, the voltage-driven diode model was utilized. An r.f. sinusoidal voltage V, sin or was applied through a
coupling capacitor and the diode admittance and r.f. power delivered by the diode into the voltage generator
Fig. 4. Calculated d.c. electric field and hole density distributions for bias current density .l, = 100 A/cm*.
621
were obtained after Fourier analysis of the diode voltage and current waveforms. The electric field was chosen as a state variable in the course of the large-signal analysis and the predictor-corrector technique was applied when solving the time derivative of the state variable. As can be seen in Fig. 4 the electric field decreases rapidly and the hole density increases very sharply inside the diode drain region. This heavily doped region, however, does not influence the microwave properties of the diode. Therefore, in order to reduce the amount of necessary computer time the large-signal analysis was confined to the part of the diode between the source contact and the drain junction. As the electric field and hole density vary smoothly within this region it was reasonable to reduce the number of space mesh points at which the state variable was originally computed from 1500 (for the d.c. analysis) to 40 (for the a.c. large-signal analysis). The mesh points were made non-equidistant with the closest spacings in the region of the source pn junction. This was because the spatial variations of electric field and carrier density are most rapid in the source region. This drastic reduction in the number of mesh points was found to save a lot of computer time yet still gave accurate results. Convergence of the predictor-corrector method to steady-state oscillations depends upon the number of time mesh points into which the period of the driving voltage is divided. It was considered that steady-state oscillations were reached when the difference between the d.c. component of the diode total current and the bias current was less than 1%. A compromise had to be chosen between the speed of convergence and the number of time mesh points to minimize the necessary computer time. About 12 to 17 cycles of the driving voltage were needed to reach the dynamic steady-state for large-signal oscillations when the number of time mesh points was 1300 per cycle. 3.
LARGE-SIGNALRESULTS
In this section we discuss the results of the large-signal analysis of the particular pnp BARITT diode the doping profile of which has already been shown in Fig. 1. The analysis was made with two main aims. These were to examine how the diode admittance, generated microwave power, efficiency and other properties depend on oscillation amplitude first when keeping the bias current constant and changing the frequency of oscillation and second when keeping the frequency constant and changing the bias current. 3.1 Fixed bias current The bias current density was fixed at J, = 150A/cm* (V. = 47 V) and the diode’s properties were investigated as a function of oscillation frequency and amplitude. In general, for very low r.f. amplitudes the injection process does not show much nonlinearity. However, when V, exceeds about 2V, injection becomes nonlinear and pulses of holes are injected when the diode terminal voltage reaches its maximum. With increasing r.f. amplitude the average diode voltage falls below the corresponding d.c. value. This effect of self-detection is shown in Fig. 5 and does not depend very much on oscillation frequency.
M. KARASEK
628
RF
VOLTAGE
AMPLITUDE
IV)
Fig 8. Power generating efficiency as a function of r.f. voltage amplitude with frequency as a parameter; J,, = 150 A/cm>.
6
IO
RF VOLTAGE
20 AMPLITUDE
IV)
Fig. 5. Average terminal voltage as a function of r.f. voltage amplitude with frequency as a parameter: J,, = IS0 A/cm*. Negative conductance appears in the frequency range between approximately 8 and 14GHz and falls sharply with r.f. amplitude as shown in Fig. 6. For V, less than 5 V the negative conductance roughly follows a I/ V,f law and the generated power is proportional to V,. The slowest fall of negative conductance occurs at about 10 GHz and results in the highest obtainable power density (P = 170 W/cm’) and efficiency (7 = 2.6%). Power density and efficiency as functions of r.f. amplitude with frequency as a parameter are plotted in Figs. 7 and 8 respectively. It was found that for the optimum frequency
-t5r
RF
VOLTAGE
AMPLITUDE
the power generation density as well as the efficiency increased with r.f. voltage until V, was approximately equal to V,,, /3. For higher r.f. amplitudes the electric field modulation was so large that, over the negative part of the terminal voltage cycle, the electric field in the vicinity of the source pn junction dropped so low that the carrier velocity was less than the saturated value across almost one half of the transit region. This results in degradation of the travelling pulses of holes owing to the effects of diffusion and in power and efficiency saturation. This saturation occurs at considerably lower r.f. amplitudes at high oscillation frequencies and at higher amplitudes for lower frequencies than the optimum frequency. The susceptance of the diode is predominatly due to its space-charge capacitance and therefore does not depend much on r.f. amplitude. Fig. 9 shows a calculated largesignal admittance plot with frequency and r.f. voltage as parameters. Although the negative conductance achieves its maximum at frequency f = I1 GHz it drops rather quickly with r.f. voltage and hence the generated power is lower than at frequency f = IO GHz. The diode’s negative Q, plotted in Fig. 10, reaches about 100 at the optimum r.f. amplitude. Dynamic distributions of hole density and electric field as functions of distance during one cycle of steady-state, large-signal oscillation are shown in Figs. 11 and 12a,b, respectively. The oscillation conditions are as follows:
IV1
Fig. 6. Negative conductance as a function of r.f. voltage amplitude with frequency as a parameter; J, = 150 A/cm*.
T 200,
,
-10
-15 RF VOLTAGE
AMPLITUDE
(VI
Fig. 7. R.F. power density as a function of r.f. voltage amplitude with frequency as a parameter: J,, = 150 A/cm’.
CONDUCTANCE
Fig. 9. Admittance
-kJ
6
Imhos/cm”l
as a function of r.f. voltage amplitude and frequency; J,, = 150 A/cm*.
629
Larse-signalanalysisof the siliconpnp-BARITTdiode
RF VOLTAGE
AMPLITUDE
IV)
Fig. 10. Negative Q as a function of r.f. voltage amplitude with frequency as a parameter; J, = 150 A/cm*.
2
4
DISTANCE
LumI
6
Fig. 11.Spatial distributions of hole density at various points in time during one ecle of large-signal steady-state J, = 150 A/cm*, f = 10 GHz, V,, = 9 V.
oscillation;
J, = 150A/cm*, f = 10GHz and V,, = 9 V. Since the average terminal voltage under this oscillation condition dropped to about 43 V and was considerably lower than the corresponding nonoscillation value the source pn junction is reverse biased until the phase angle. I#J= wt of the driving voltage reaches approximately 30”. By this time the pulse of holes from the previous oscillation cycle has already been absorbed into the heavily doped drain region. As the phase angle passes 60” holes are still injected into the transit region. When 4 exceeds 90” the terminal voltage begins to decrease and owing to the screening effect of the already injected charge the source pn junction becomes reverse biased and injection ceases. Because the injection takes place in a short time interval (30”~ 4
6
I
0
L-I_._
2
4
DISTANCE
(pm)
Fig. 12a,b.Spatialdistributionsof electricfieldat variouspointsin time during one cycle of large signal steady state oscillation; J, = 150 A/cm*,f = 10 GHz, V, = 9 V.
SSE Vol. 19. No. 1-E
DISTANCE
(Nm)
Fig. 13. Dynamic distributions of hole density in the vicinity of the source pn junction; J, = 150 A/cm*, f = 10 GHz. V,, = 9 V.
630
M. KARASEK
105 1 ‘6
L RE
Fig. 17. Susceptance
20
lb
VOLTAGE
AMPLITUDE
(VI
vs r.f. voltage amplitude with bias current = 10 GHz.
density as aparameter;f RF
VOLTAGE
AMPLITUDE
Fig. 14. Power generating efficiency vs r.f. voltage amplitude with bias current density as a parameter; f = 10 GHz.
io RF
VOLTAGE
’
io
AMPLITUDE
I (‘4
Fig. IS. R.F. power density vs r.f. voltage amplitude with bias current density as a parameter; f = 10 GHz.
RF VOLTAGE
AMPLITUDE
(VI
Fig. 16. Negative conductance vs r.f. voltage amplitude with bias current density as a parameter; f = IO GHz.
and 17, respectively. The change of susceptance with bias current, the electronic tuning effect, is larger at small r.f. voltage amplitudes than at larger ones. A comprehensive view of the large-signal admittance behaviour is given in Fig. 18, where the admittance of the diode is plotted as a function of r.f. voltage amplitude and bias current density. This electronic tuning effect suggests possible uses for the device as an amplitude or frequency modulated source. 4. CONCLUSIONS
The mathematical model which has been outlined for the BARITT diode with two pn junctions clarifies the
L
v
J,=2OOA/cm*
-Ib CONDUCTANCE
*IO5 4
2,
(mhos/cm*l
Fig. 18. Admittance as a function of r.f. voltage amplitude and bias current density; f = 10 GHz.
electronic and physical mechanisms of operation and provides much realistic information on the device largesignal behaviour. This model takes into account all the necessary physical conditions such as a realistic doping profile, carrier velocity-electric field relation, and both drift and diffusion components of particle current. Because of the reduction of the number of space mesh points during the large-signal analysis it was possible to reduce the necessary computer time to a reasonable value. Dynamic space distributions of the electric field and hole density have been presented at various points in time during one cycle of steady-state oscillation. The diode’s conductance, susceptance, power density and efficiency have been presented as functions of oscillation amplitude, frequency and bias current density. It has been shown that the pnp BARITT diode is fundamentally of low efficiency but even so can generate sufficient power for many practical applications such as a.m. and f.m. sources, local oscillators, self-oscillating mixers, doppler radars and communications and broadcasting receivers. Its potential use for such applications is enhanced by its stability and reliability of operation and by the fact that its fabrication is based on silicon technology. Acknowledgements--It is a pleasure to thank G. T. Wright for many helpful discussions and to acknowledge the award of a Leverhulme Trust Fellowship for 1974/75.
Large-signalanalysis of the silicon pnp-BARITT diode REFERENCFS
1. G. T. Wright, Elect. Letr. 4, 543 (1968). 2. H. W. Ruegg, I.E.E.E. Trans. ED-IS, 577 (1%8). 3. N. B. Sultan and G. T. Wright, U.K. Ministry of Tech. Report, AT/2027/061/RL(1970). 4. N. B. Sultan and G. T. Wrieht. Elect. Lett. 8. 24 (1972). 5. D. J. Coleman and S. M. Sze; Bell Syst. Tech. i 50,’ 1695 (1971). 6. S. G. Liu and J. J. Risko, RCA Rev. 32, 636 (1971). 7. C. P. Snapp and P. Weissglass, Elect. Lett. 7, 743 (1971). 8. J. L. Fikart, 1.E.E.E. Trans. MTT 22, 517 (1974). 9. G. T. Wright, Solid-St. Electron. 16, 903 (1973).
631
10. E. P. EerNise, Appl. Phys. Lett. 20, 301 (1972). 11. K. P. Weller, RCA Rev. 32, 372 (1971). 12. J. A. C. Stewart, Elect. Lett. 10, 193 (1974). 13. M. Matsumura, Trans. Inst. Electron. Comm. Engin., Japan 56C, 242 (1973).
14. U. B. Sheorey, I. Lundstrom and E. A. Ash, In. J. Electron. 30, 19 (1971). 15. G. T. Wright, Solid-St. Electron. 19, 615 (1976). 16. A. DeMari, Solid-St. Electron. 11, 3 (1%8). 17. D. M. Caughey and B. E. Thomas, Proc. I.E.E.E. 55, 2192 (1%7).
Electronics. 1976. Vol. 19. pp. 625-631.
Pergamon Pres.
Prmted in Great Bntaio
LARGE-SIGNAL ANALYSIS OF THE SILICON pnp -BARITT DIODE M. KARASEKt Department of Electronic and Electrical Engineering, University of Birmingham,BirminghamB152TT, England (Received 31 October 1975;in
revisedfonu
23 January
1976)
Abstract-A numerical large-signal analysis of the silicon pnp BARITT diode is presented. This extends previous works to practical device structures. The diode admittance, power generation density, efficiency and quality factor are studied as functions of oscillation amplitude, frequency and bias current density. The results are found to be in substantial agreement with large-signal experimental measurements. A typical pup structure optimised for operation at 10GHz has an active region doping density of 2 x lO”/cm’, a width of 5.5 pm and a punch-through voltage of 40 V. Negative resistance occurs over the frequency range from about 8 GHz to about 14GHz. The device Q at 10GHz varies from a small-signal value of about -10 to a large-signal value of about -100. The conversion efficiency at IOGHz is about 5% at a bias current density of 25 A/cm* and falls to about 2% at 200A/cm*. Maximum r.f. power outnut occurs at an r.f. amolitude of about 15V and rises with bias current density to a maximum value of about 176W/cm2at 200A/cm’. L 1. INTRODUCTION
1%8 Wright[l] described a new negative resistance microwave device based on the principle of barrier controlled injection and transit time delay-the BARITT diode. His simple analysis suggested that the device should operate at moderate power and low noise levels. In the same year, independently, Ruegg presented a paper [2] on the simplified large-signal theory of a similar punchthrough structure giving considerably optimistic prospects-an estimated efficiency of the order of 20% and power output of 10-100 W at 10GHz. These theoretical works were confirmed experimentally in 1970 when Sultan and Wright [3] achieved negative resistance in npn silicon structures, and subsequently oscillations in pnp structures[4] and in 1971 when Coleman and Sze[S] reported oscillations in metal-semiconductor-metal structures. Several experimental papers have since been presented [6,7,8] comparing the properties of different BARIn diode structures and pointing out the reliable and low-noise operation of the device at moderate power levels. Following the achievement of microwave oscillations, extensive theoretical analyses and numerical calculations have been reported discussing the small-signal behaviour of various BARITT structures [9,10,11] and providing useful insight into the physical and electronic mechanisms of their operation. However, for utilization of the BARITT diode as a free-running oscillator, modulated source or self-oscillating mixer the knowledge of the large-signal properties is of vital importance. Unfortunately only a few contributions are available on the large-signal analysis of BARITT diodes[2, 12-141. The main reason seems to be the difficulty of solving the nonlinear system of partial differential equations governing carrier injection and transport. Even if a onedimensional model is used, it is impossible to obtain a In
closed analytical solution without severe simplifications such as space-charge-limited injection, neglect of carrier diffusion and the assumption of saturated carrier velocity throughout the whole active region. Such an analysis has been made by Wright[lS] and can give useful and informative first-order results. The aim of this paper is to present a numerical technique which has been used successfully for the large-signal analysis of the BARITT diode with two pn junctions and which takes into account all the necessary practical physical conditions such as a realistic doping profile, a suitable carrier velocity-electric field relation, and both drift and diffusion components of particle current. The computer programme which has been developed has been applied to a typical X-band pnp silicon BARITT diode and large-signal values of admittance, power generation density and efficiency are shown to be dependent on oscillation amplitude, frequency and bias current density. The time and space dependence of the electric field intensity and positive hole density in the case of largesignal oscillations have also been studied. Despite the relative complexity of the basic equations the programme is not extremely computer time consuming and may be used for parameter optimization of pnp, pnup and similar BARITT structures with two pn junctions. 2. MATHEMATICAL MODEL
Although the mathematical descriptions of the processes inside different BARITT diode structures are very similar we restrict ourselves in the following analysis to a pnp structure. This is the most basic design and is the most practical for device fabrication. In the active state the central transit region of this structure is completely depleted and the transport of injected carriers is described by Poisson’s equation, the continuity equation and Maxwell’s equation of total current together with the equation of particle current density. These equations are as follows
ton leavefrom Institute of RadioEngineeringand Electronics, CzechoslovakAcademy of Sciences, 180 88 Praha, Czechoslovakia.
dE(x,t) dX
625
e
=T[p(x,t)-NA(x)+No(x)l
(1)
M. KARASEK
626
G,(x) g [PdX)P(X)~
t)v
A&
[2-pK(Ax)*]Ai,+A$K+I=A
I-
where E X f! E
P NA, ND t J, JT l-h D,
BpwtCpK+Dp,c+I=O
electric field intensity, space coordinate, electronic charge, permittivity of semiconductor material, positive hole density, density of acceptor and donor impurities, time, hole current density, total current density hole mobility, hole diffusion coefficient.
-
(pK
+ pK
,,)/2
exp
K = 2,3,4. .
ax
J,(x,t)=p,(x,f)
pkt)E(x,t)-jy-
aP(x, t)
I
(8)
The diffusion constant in eqn (4) was substituted in terms of the carrier mobility by means of the well known Einstein relation.? The Caughey-Thomas formulae [ 171 have been used for the carrier velocity-electric field relations. To complete the specification of the mathematical model a set of initial and boundary conditions must be defined. The electric field spatial distribution corresponding to a given d.c. bias is taken as the initial condition for the subsequent large-signal analysis. When computing the d.c. field distribution the original set of partial differential equations is converted into two ordinary differential equations as follows dE(x) dx
-=p(x)+
+This is justified because importantin thoseregionsandat small.
(h
-
4K+d
1499.
(13)
The above sets are solved iteratively together with boundary conditions which, under the assumption of infinite recombination contact regions, are in the form E(x,t)=O p(x,t)=N(x)
__ aJ,(~~t)
at
(12)
D =(/.LcLK +pKtl)/2exp(rlrK-l-IlrK)
=tx, t) -=p(x,t)+N(x) ax t)_
(11)
where AeK is a correction to the electrostatic potential. The coefficients A, B, C, and D are defined as follows:
In order to simplify the mathematical manipulations and to avoid possible numerical overflows and underflows during execution of the programme it is convenient to normalize the above set of equations into a dimensionless form. The normalization factors used are the same as those proposed by DeMari [ 161.When keeping the same symbols the normalized equations are
a~0 A_
(10)
where (p,, is the quasi-Fermi level for holes. These nonlinear equations are linearized by the NewtonRaphson method and the usual finite difference technique is applied leading to two sets of linear algebraic equations
40, t) = wpk t)p(x, t)Ek f) -eD,(x,
I= 0
x =O,L.
(14)
In order to demonstrate the calculation of initial conditions several results of the d.c. analysis are presented. A typical pnp device structure was chosen designed for operation at X-band. This is shown in Fig. 1. The transit region was 5.5 pm wide and was uniformly doped with a donor density of ND = 2 x lOI5cm-‘. The calculated d.c. current-voltage characteristic of the diode is shown in Fig. 2. When the voltage is below the punch-through value ( V,, = 39 V) there is a sufficiently high potential barrier in front of the source pn junction that only a small amount of holes is injected into the transit region and almost no current flows. The spatial distributions of hole density and electrostatic potential in the vicinity of the source pn junction are displayed in Fig. 3. When the bias voltage
N(x)
the effects of diffusion are only those timeswhen theelectricfieldis
DISTANCE
iflrnl
Fig. I. Doping concentration profile used for the silicon pnp BARITT diode.
Large-signal analysis of the silicon pnp-BARITT diode
40;
i 0
__I._ 50 CURRENT
100DENSITY
--x7 (A/cm’)
200
Fig. 2. Calculated dc. current-voltage characteristic.
POTENTIAL
Y 0 I
1
L.00,
\,
IO8
Fig. 3. Distribution of d.c. hole density and electrostatic potential in the vicinity of source pn junction.
exceeds 39V, the potential barrier of the source pn junction begins to diminish, the amount of injected holes increases and the diode current sharply rises. Figure 4 shows the static spatial distributions of the electric field and hole density throughout the structure corresponding to a d.c. current density J, = 100A/cm2. In order to study the large-signal performance of the BARITT diode at a given frequency but for various bias conditions, the voltage-driven diode model was utilized. An r.f. sinusoidal voltage V, sin or was applied through a
coupling capacitor and the diode admittance and r.f. power delivered by the diode into the voltage generator
Fig. 4. Calculated d.c. electric field and hole density distributions for bias current density .l, = 100 A/cm*.
621
were obtained after Fourier analysis of the diode voltage and current waveforms. The electric field was chosen as a state variable in the course of the large-signal analysis and the predictor-corrector technique was applied when solving the time derivative of the state variable. As can be seen in Fig. 4 the electric field decreases rapidly and the hole density increases very sharply inside the diode drain region. This heavily doped region, however, does not influence the microwave properties of the diode. Therefore, in order to reduce the amount of necessary computer time the large-signal analysis was confined to the part of the diode between the source contact and the drain junction. As the electric field and hole density vary smoothly within this region it was reasonable to reduce the number of space mesh points at which the state variable was originally computed from 1500 (for the d.c. analysis) to 40 (for the a.c. large-signal analysis). The mesh points were made non-equidistant with the closest spacings in the region of the source pn junction. This was because the spatial variations of electric field and carrier density are most rapid in the source region. This drastic reduction in the number of mesh points was found to save a lot of computer time yet still gave accurate results. Convergence of the predictor-corrector method to steady-state oscillations depends upon the number of time mesh points into which the period of the driving voltage is divided. It was considered that steady-state oscillations were reached when the difference between the d.c. component of the diode total current and the bias current was less than 1%. A compromise had to be chosen between the speed of convergence and the number of time mesh points to minimize the necessary computer time. About 12 to 17 cycles of the driving voltage were needed to reach the dynamic steady-state for large-signal oscillations when the number of time mesh points was 1300 per cycle. 3.
LARGE-SIGNALRESULTS
In this section we discuss the results of the large-signal analysis of the particular pnp BARITT diode the doping profile of which has already been shown in Fig. 1. The analysis was made with two main aims. These were to examine how the diode admittance, generated microwave power, efficiency and other properties depend on oscillation amplitude first when keeping the bias current constant and changing the frequency of oscillation and second when keeping the frequency constant and changing the bias current. 3.1 Fixed bias current The bias current density was fixed at J, = 150A/cm* (V. = 47 V) and the diode’s properties were investigated as a function of oscillation frequency and amplitude. In general, for very low r.f. amplitudes the injection process does not show much nonlinearity. However, when V, exceeds about 2V, injection becomes nonlinear and pulses of holes are injected when the diode terminal voltage reaches its maximum. With increasing r.f. amplitude the average diode voltage falls below the corresponding d.c. value. This effect of self-detection is shown in Fig. 5 and does not depend very much on oscillation frequency.
M. KARASEK
628
RF
VOLTAGE
AMPLITUDE
IV)
Fig 8. Power generating efficiency as a function of r.f. voltage amplitude with frequency as a parameter; J,, = 150 A/cm>.
6
IO
RF VOLTAGE
20 AMPLITUDE
IV)
Fig. 5. Average terminal voltage as a function of r.f. voltage amplitude with frequency as a parameter: J,, = IS0 A/cm*. Negative conductance appears in the frequency range between approximately 8 and 14GHz and falls sharply with r.f. amplitude as shown in Fig. 6. For V, less than 5 V the negative conductance roughly follows a I/ V,f law and the generated power is proportional to V,. The slowest fall of negative conductance occurs at about 10 GHz and results in the highest obtainable power density (P = 170 W/cm’) and efficiency (7 = 2.6%). Power density and efficiency as functions of r.f. amplitude with frequency as a parameter are plotted in Figs. 7 and 8 respectively. It was found that for the optimum frequency
-t5r
RF
VOLTAGE
AMPLITUDE
the power generation density as well as the efficiency increased with r.f. voltage until V, was approximately equal to V,,, /3. For higher r.f. amplitudes the electric field modulation was so large that, over the negative part of the terminal voltage cycle, the electric field in the vicinity of the source pn junction dropped so low that the carrier velocity was less than the saturated value across almost one half of the transit region. This results in degradation of the travelling pulses of holes owing to the effects of diffusion and in power and efficiency saturation. This saturation occurs at considerably lower r.f. amplitudes at high oscillation frequencies and at higher amplitudes for lower frequencies than the optimum frequency. The susceptance of the diode is predominatly due to its space-charge capacitance and therefore does not depend much on r.f. amplitude. Fig. 9 shows a calculated largesignal admittance plot with frequency and r.f. voltage as parameters. Although the negative conductance achieves its maximum at frequency f = I1 GHz it drops rather quickly with r.f. voltage and hence the generated power is lower than at frequency f = IO GHz. The diode’s negative Q, plotted in Fig. 10, reaches about 100 at the optimum r.f. amplitude. Dynamic distributions of hole density and electric field as functions of distance during one cycle of steady-state, large-signal oscillation are shown in Figs. 11 and 12a,b, respectively. The oscillation conditions are as follows:
IV1
Fig. 6. Negative conductance as a function of r.f. voltage amplitude with frequency as a parameter; J, = 150 A/cm*.
T 200,
,
-10
-15 RF VOLTAGE
AMPLITUDE
(VI
Fig. 7. R.F. power density as a function of r.f. voltage amplitude with frequency as a parameter: J,, = 150 A/cm’.
CONDUCTANCE
Fig. 9. Admittance
-kJ
6
Imhos/cm”l
as a function of r.f. voltage amplitude and frequency; J,, = 150 A/cm*.
629
Larse-signalanalysisof the siliconpnp-BARITTdiode
RF VOLTAGE
AMPLITUDE
IV)
Fig. 10. Negative Q as a function of r.f. voltage amplitude with frequency as a parameter; J, = 150 A/cm*.
2
4
DISTANCE
LumI
6
Fig. 11.Spatial distributions of hole density at various points in time during one ecle of large-signal steady-state J, = 150 A/cm*, f = 10 GHz, V,, = 9 V.
oscillation;
J, = 150A/cm*, f = 10GHz and V,, = 9 V. Since the average terminal voltage under this oscillation condition dropped to about 43 V and was considerably lower than the corresponding nonoscillation value the source pn junction is reverse biased until the phase angle. I#J= wt of the driving voltage reaches approximately 30”. By this time the pulse of holes from the previous oscillation cycle has already been absorbed into the heavily doped drain region. As the phase angle passes 60” holes are still injected into the transit region. When 4 exceeds 90” the terminal voltage begins to decrease and owing to the screening effect of the already injected charge the source pn junction becomes reverse biased and injection ceases. Because the injection takes place in a short time interval (30”~ 4
6
I
0
L-I_._
2
4
DISTANCE
(pm)
Fig. 12a,b.Spatialdistributionsof electricfieldat variouspointsin time during one cycle of large signal steady state oscillation; J, = 150 A/cm*,f = 10 GHz, V, = 9 V.
SSE Vol. 19. No. 1-E
DISTANCE
(Nm)
Fig. 13. Dynamic distributions of hole density in the vicinity of the source pn junction; J, = 150 A/cm*, f = 10 GHz. V,, = 9 V.
630
M. KARASEK
105 1 ‘6
L RE
Fig. 17. Susceptance
20
lb
VOLTAGE
AMPLITUDE
(VI
vs r.f. voltage amplitude with bias current = 10 GHz.
density as aparameter;f RF
VOLTAGE
AMPLITUDE
Fig. 14. Power generating efficiency vs r.f. voltage amplitude with bias current density as a parameter; f = 10 GHz.
io RF
VOLTAGE
’
io
AMPLITUDE
I (‘4
Fig. IS. R.F. power density vs r.f. voltage amplitude with bias current density as a parameter; f = 10 GHz.
RF VOLTAGE
AMPLITUDE
(VI
Fig. 16. Negative conductance vs r.f. voltage amplitude with bias current density as a parameter; f = IO GHz.
and 17, respectively. The change of susceptance with bias current, the electronic tuning effect, is larger at small r.f. voltage amplitudes than at larger ones. A comprehensive view of the large-signal admittance behaviour is given in Fig. 18, where the admittance of the diode is plotted as a function of r.f. voltage amplitude and bias current density. This electronic tuning effect suggests possible uses for the device as an amplitude or frequency modulated source. 4. CONCLUSIONS
The mathematical model which has been outlined for the BARITT diode with two pn junctions clarifies the
L
v
J,=2OOA/cm*
-Ib CONDUCTANCE
*IO5 4
2,
(mhos/cm*l
Fig. 18. Admittance as a function of r.f. voltage amplitude and bias current density; f = 10 GHz.
electronic and physical mechanisms of operation and provides much realistic information on the device largesignal behaviour. This model takes into account all the necessary physical conditions such as a realistic doping profile, carrier velocity-electric field relation, and both drift and diffusion components of particle current. Because of the reduction of the number of space mesh points during the large-signal analysis it was possible to reduce the necessary computer time to a reasonable value. Dynamic space distributions of the electric field and hole density have been presented at various points in time during one cycle of steady-state oscillation. The diode’s conductance, susceptance, power density and efficiency have been presented as functions of oscillation amplitude, frequency and bias current density. It has been shown that the pnp BARITT diode is fundamentally of low efficiency but even so can generate sufficient power for many practical applications such as a.m. and f.m. sources, local oscillators, self-oscillating mixers, doppler radars and communications and broadcasting receivers. Its potential use for such applications is enhanced by its stability and reliability of operation and by the fact that its fabrication is based on silicon technology. Acknowledgements--It is a pleasure to thank G. T. Wright for many helpful discussions and to acknowledge the award of a Leverhulme Trust Fellowship for 1974/75.
Large-signalanalysis of the silicon pnp-BARITT diode REFERENCFS
1. G. T. Wright, Elect. Letr. 4, 543 (1968). 2. H. W. Ruegg, I.E.E.E. Trans. ED-IS, 577 (1%8). 3. N. B. Sultan and G. T. Wright, U.K. Ministry of Tech. Report, AT/2027/061/RL(1970). 4. N. B. Sultan and G. T. Wrieht. Elect. Lett. 8. 24 (1972). 5. D. J. Coleman and S. M. Sze; Bell Syst. Tech. i 50,’ 1695 (1971). 6. S. G. Liu and J. J. Risko, RCA Rev. 32, 636 (1971). 7. C. P. Snapp and P. Weissglass, Elect. Lett. 7, 743 (1971). 8. J. L. Fikart, 1.E.E.E. Trans. MTT 22, 517 (1974). 9. G. T. Wright, Solid-St. Electron. 16, 903 (1973).
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10. E. P. EerNise, Appl. Phys. Lett. 20, 301 (1972). 11. K. P. Weller, RCA Rev. 32, 372 (1971). 12. J. A. C. Stewart, Elect. Lett. 10, 193 (1974). 13. M. Matsumura, Trans. Inst. Electron. Comm. Engin., Japan 56C, 242 (1973).
14. U. B. Sheorey, I. Lundstrom and E. A. Ash, In. J. Electron. 30, 19 (1971). 15. G. T. Wright, Solid-St. Electron. 19, 615 (1976). 16. A. DeMari, Solid-St. Electron. 11, 3 (1%8). 17. D. M. Caughey and B. E. Thomas, Proc. I.E.E.E. 55, 2192 (1%7).