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Current instabilities in “s-shaped” negative differential conductivity elements
JOURNAL OF NoN-CRYSTALLINE SOLIDS 8-10 (1972) 999-1003 © North-Holland Publishing Co.
C U R R E N T I N S T A B I L I T I E S I N "S-SHAPED" NEGATIVE DIFFERENTIAL CONDUCTIVITY ELEMENTS M. P. SHAW and I. J. GASTMAN Department of Electrical Engineering, Wayne State University, Detroit, Michigan 48202, U.S.A.
We have elucidated the important reactive components of"S-shaped" negative differential conductivity (NDC) elements and established the form and nature of the current instabilities. Our experimental results for PNPN and Ovonic diodes concur with the approximations we have made; all the major features of the experimental results are accounted for by the theory. The theoretical results can be directly transformed between S and Nshaped NDC elements (e.g., tunnel and Gunn diodes) and provide a basis within which a general theory for an arbitrarily shaped conduction current curve can be established. We have addressed ourselves to the problem of understanding the large signal behavior of an inhomogeneous material exhibiting negative differential conductivity (NDC), under the influence of an externally applied dc electric field, where the field is imposed by attaching conducting leads to the material. When N D C is present sustained relaxation oscillations can be generated. In a lumped element approximation our model for the important circuit parameters in the local environment of the material is shown in fig. l a, where we have replaced the inhomogeneous material by a non-linear resistor having a conduction current curve VD(i¢) in parallel with an intrinsic capacitor and in series with an intrinsic and/or package inductor. VD(ic) generally exhibits hysteresis during an oscillatory cycle. The shape of VD(ic) is determined by the field or current density evolution within the N D C element, which depends on the boundary conditions, homogeneity, and circuit parameters. Our goal has been to categorize and characterize the various modes of behavior possible in this environment. Solomon, Grubin and Shaw 1) have recently made substantial progress in understanding the problem for N-shaped N D C elements, such as the tunnel diode, where the internal fields vary uniformly, and the Gunn diode, which is unstable against high field domain formation. In this paper we discuss our progress in the study of S-shaped N D C elements, such as the P N P N diode, where the current density varies uniformly, and the Ovonic diode, which is unstable against high current density filament formation. We have discovered that when the important reactive components are iden999
1000
M. P. S H A W A N D I. J. G A S T M A N
tiffed for N and S-shaped NDC elements, the theory transforms between the two with only minor modification. We have found the so called "dual" circuit 2) for the system. As shown in fig. la, there are four important reactive components present in our model of the local environment. Thus, the nonlinear differential equation governing the temporal response of the conduction current ic(t ) will be of fourth order. The equation can be reduced to second order3), however, by realizing that in the time dependent solutions for S-shaped NDC elements, the amplitude of the intrinsic displacement current is much less than the conduction current and the amplitude of the voltage drop across the load resistor is much greater than the induced voltage across the lead inductance. The problem then reduces to what we refer to as the "primary" circuit shown in fig. lb. Our experimental results concur with this approximation. We obtain approximate analytic solutions for the total voltage V across the NDC element plus intrinsic inductor, as a function of io by piecewise linearizing the S-shaped conduction current curve Vo(ic). An example of a nonlinear curve for uniform current density is shown in fig. 2a. Here the V(ic) trajectory in the relaxation oscillation mode is superimposed on the S-shaped curve. We have recently presented criteria for the estblishament of relaxation oscillations in this system 3). jPI
R
1
040|0
L~
L,
T I
(a)
C
(b)
Fig. 1. (a) The conduction current characteristic Vn(io) of an S-shaped N D C element, and its local environment. Battery voltage VB, load resistor R, lead inductance LL, package capacitance Cp, intrinsic and/or package inductance Ll, intrinsic capacitance Cl. The contact resistance is included in V9(ic). (b) Circuit under analysis (primary circuit).
1001
CURRENT INSTABILITIES
4"
V
V"
.
Ic (a)
(bl
ic
Fig. 2. (a) Voltage versus conduction current. When the dc load line (dV/dt=O) intersects the conduction current curve Vo(ie) (heavy) at a point in the NDC region (×) and the conditions for sustained relaxation oscillations are met3), a steady state trajectory exemplified by V(ie) occurs. VD(ie) has three regions: (I) low current (OFF); (If) NDC; (III) high current (ON). lid (ic) exhibits hysteresis when the current density becomes nonuniform, but this is not shown for the purpose of clarity. (b) Experimental data (traced from a photograph taken from the face of a Tektronix 564 storage oscilloscope); a typical V(iO curve for a PNPN diode. Vertical, 2 V/div. Horizontal, 2 mA/div. The dark circle is the origin.
In order to verify the results o f our analysis we have studied several M o t o r o l a M 4 L "four-layer" P N P N diodes and Ovonic DO-7 switching diodes4). The experimental details are described in ref. 3. At r o o m temperature the P N P N diode is stable in its region o f negative slope and is made unstable by applying an external capacitor in parallel. Thus, by inserting a small current monitoring resistor inside the external capacitor and in series with the diode, the V(ic) trajectory shown in fig. 2a can be obtained experimentally. A typical example is shown in fig. 2b. It compares favorably with the theoretical curve. The V(ic) trajectory is clearly experimentally accessible in devices where the conduction current can be measured. Since the Ovonic diode is unstable in its N D C region against filament formation 5) and can be induced into relaxation oscillations even in the absence o f an external capacitor, the conduction current is not readily attainable. In order to compare theory with experiment for these devices we must therefore first transform the theoretical trajectory to the total current-voltage plane. I f the abscissa in fig. 2a is changed to the total current I( = i¢ + Cd V/dt), the trajectory collapses to the load line. In this plane the voltage extrema are also the total current extrema. In ref. 3
1002
M. P. S H A W A N D 1. J. G A S T M A N
we have displayed representative total current-voltage data for a P N P N diode shunted by an external capacitor, and an Ovonic diode. The data again compare favorably with the theory. The fact that the experimental oscillations appear as a straight line along the load line (although sometimes they exhibit a slight "bowing") is evidence that our approximation of neglecting the lead inductance is reasonable. If the lead inductance were important the oscillation line would appear as a closed loop. In the Ovonic diode, as the battery voltage is increased above threshold in the relaxation oscillation regime the extrema points on the oscillation line approach each other a). When the extremum point corresponding to minimum voltage becomes too high, however, switching occurs to a voltage near the holding voltage. This behavior is expected for the following reasons. When the current density becomes inhomogenous (filament formation), the conduction current and NDC element voltage mirror this and become time dependent. VD(ic) then exhibits hysteresis during an oscillatory cycle 6). As the battery voltage is increased the threshold voltage decreases and the minimum conduction current reached each cycle increases 6). When it exceeds a "filament sustaining current" [analogue of "domain sustaining field" 1)] the filament is not quenched at the end of the cycle and switching occurs to a voltage near the holding voltage. V(ic)"spirals in" to the switched or ON state6). We find that a spiral approach to the ON state is fundamental to the switching process in general. This behavior is discussed in detail for S-shaped N D C elements in ref. 1, where we make use of an "n subelement" model for inhomogeneous S-shaped N D C elements. Here we model the inhomogenous material by a parallel array of S-shaped conduction current curves. The S-shaped curves have various values of voltage "peak to valley" ratios. In summary, we review the major points and conclusions of our work. First, we have performed a large signal analysis of an arbitrary "S-shaped" N D C element placed in a local environment containing the vital reactive components involved with such an element; the primary circuit. The results provide a firm foundation for the understanding of a wide variety of current instabilities generated in PNPN and Ovonic diodes. To our knowledge, ours is the first detailed large signal circuit analysis applied to an S-shaped N D C element. Our experimental results concur with the approximations we have made. Second, we have found that our theory describes the behavior of both N and S-shaped N D C elements when the important reactive components are identified for each case. For the N-shaped case the primary circuit has the inductor outside the capacitor1), that is, the intrinsic inductance is ignored. The differential equation for the voltage across the N D C element in this case is the same as the equation for the conduction current in the S-shaped case. In other words, we have found that the important circuit parameters in each
CURRENT INSTABILITIES
1003
case p r o d u c e a " d u a l " circuit system where the voltage across the N - s h a p e d N D C element in its p r i m a r y circuit behaves exactly as the current t h r o u g h the S - s h a p e d N D C element in its p r i m a r y circuit. Therefore, our circuit t h e o r y o f a G u n n d i o d e can be t r a n s f o r m e d directly to the case o f an O v o n i c diode, where the high c u r r e n t density filament is the a n a l o g u e o f a high field d o m a i n . W e have benefited f r o m discussions with P. R. S o l o m o n , H. L. G r u b i n , D. A d l e r , E. A. Fagen, a n d H. Fritzsche. W e t h a n k W. Evans for considerable assistance in t a k i n g the data.
References 1) M. P. Shaw, P. R. Solomon and H. L. Grubin, Appl. Phys. Letters 17 (1970) 535; P. R. Solomon, M. P. Shaw and H. L. Grubin, J. Appl. Phys. 43 (1972) 159. 2) This feature was pointed out by G. B. Yntema. 3) M. P. Shaw and I. J. Gastman, Appl. Phys. Letters 19 (1971) 243. 4) Kindly provided by R. F. Shaw at ECD, Inc. 5) See, e.g., K, Homma, Appl. Phys. Letters 18 (1971) 198. 6) M. P. Shaw, H. L. Grubin and I. J. Gastman, IEEE Trans. Electron. Devices, Special issue on amorphous semiconductors, to be published.
C U R R E N T I N S T A B I L I T I E S I N "S-SHAPED" NEGATIVE DIFFERENTIAL CONDUCTIVITY ELEMENTS M. P. SHAW and I. J. GASTMAN Department of Electrical Engineering, Wayne State University, Detroit, Michigan 48202, U.S.A.
We have elucidated the important reactive components of"S-shaped" negative differential conductivity (NDC) elements and established the form and nature of the current instabilities. Our experimental results for PNPN and Ovonic diodes concur with the approximations we have made; all the major features of the experimental results are accounted for by the theory. The theoretical results can be directly transformed between S and Nshaped NDC elements (e.g., tunnel and Gunn diodes) and provide a basis within which a general theory for an arbitrarily shaped conduction current curve can be established. We have addressed ourselves to the problem of understanding the large signal behavior of an inhomogeneous material exhibiting negative differential conductivity (NDC), under the influence of an externally applied dc electric field, where the field is imposed by attaching conducting leads to the material. When N D C is present sustained relaxation oscillations can be generated. In a lumped element approximation our model for the important circuit parameters in the local environment of the material is shown in fig. l a, where we have replaced the inhomogeneous material by a non-linear resistor having a conduction current curve VD(i¢) in parallel with an intrinsic capacitor and in series with an intrinsic and/or package inductor. VD(ic) generally exhibits hysteresis during an oscillatory cycle. The shape of VD(ic) is determined by the field or current density evolution within the N D C element, which depends on the boundary conditions, homogeneity, and circuit parameters. Our goal has been to categorize and characterize the various modes of behavior possible in this environment. Solomon, Grubin and Shaw 1) have recently made substantial progress in understanding the problem for N-shaped N D C elements, such as the tunnel diode, where the internal fields vary uniformly, and the Gunn diode, which is unstable against high field domain formation. In this paper we discuss our progress in the study of S-shaped N D C elements, such as the P N P N diode, where the current density varies uniformly, and the Ovonic diode, which is unstable against high current density filament formation. We have discovered that when the important reactive components are iden999
1000
M. P. S H A W A N D I. J. G A S T M A N
tiffed for N and S-shaped NDC elements, the theory transforms between the two with only minor modification. We have found the so called "dual" circuit 2) for the system. As shown in fig. la, there are four important reactive components present in our model of the local environment. Thus, the nonlinear differential equation governing the temporal response of the conduction current ic(t ) will be of fourth order. The equation can be reduced to second order3), however, by realizing that in the time dependent solutions for S-shaped NDC elements, the amplitude of the intrinsic displacement current is much less than the conduction current and the amplitude of the voltage drop across the load resistor is much greater than the induced voltage across the lead inductance. The problem then reduces to what we refer to as the "primary" circuit shown in fig. lb. Our experimental results concur with this approximation. We obtain approximate analytic solutions for the total voltage V across the NDC element plus intrinsic inductor, as a function of io by piecewise linearizing the S-shaped conduction current curve Vo(ic). An example of a nonlinear curve for uniform current density is shown in fig. 2a. Here the V(ic) trajectory in the relaxation oscillation mode is superimposed on the S-shaped curve. We have recently presented criteria for the estblishament of relaxation oscillations in this system 3). jPI
R
1
040|0
L~
L,
T I
(a)
C
(b)
Fig. 1. (a) The conduction current characteristic Vn(io) of an S-shaped N D C element, and its local environment. Battery voltage VB, load resistor R, lead inductance LL, package capacitance Cp, intrinsic and/or package inductance Ll, intrinsic capacitance Cl. The contact resistance is included in V9(ic). (b) Circuit under analysis (primary circuit).
1001
CURRENT INSTABILITIES
4"
V
V"
.
Ic (a)
(bl
ic
Fig. 2. (a) Voltage versus conduction current. When the dc load line (dV/dt=O) intersects the conduction current curve Vo(ie) (heavy) at a point in the NDC region (×) and the conditions for sustained relaxation oscillations are met3), a steady state trajectory exemplified by V(ie) occurs. VD(ie) has three regions: (I) low current (OFF); (If) NDC; (III) high current (ON). lid (ic) exhibits hysteresis when the current density becomes nonuniform, but this is not shown for the purpose of clarity. (b) Experimental data (traced from a photograph taken from the face of a Tektronix 564 storage oscilloscope); a typical V(iO curve for a PNPN diode. Vertical, 2 V/div. Horizontal, 2 mA/div. The dark circle is the origin.
In order to verify the results o f our analysis we have studied several M o t o r o l a M 4 L "four-layer" P N P N diodes and Ovonic DO-7 switching diodes4). The experimental details are described in ref. 3. At r o o m temperature the P N P N diode is stable in its region o f negative slope and is made unstable by applying an external capacitor in parallel. Thus, by inserting a small current monitoring resistor inside the external capacitor and in series with the diode, the V(ic) trajectory shown in fig. 2a can be obtained experimentally. A typical example is shown in fig. 2b. It compares favorably with the theoretical curve. The V(ic) trajectory is clearly experimentally accessible in devices where the conduction current can be measured. Since the Ovonic diode is unstable in its N D C region against filament formation 5) and can be induced into relaxation oscillations even in the absence o f an external capacitor, the conduction current is not readily attainable. In order to compare theory with experiment for these devices we must therefore first transform the theoretical trajectory to the total current-voltage plane. I f the abscissa in fig. 2a is changed to the total current I( = i¢ + Cd V/dt), the trajectory collapses to the load line. In this plane the voltage extrema are also the total current extrema. In ref. 3
1002
M. P. S H A W A N D 1. J. G A S T M A N
we have displayed representative total current-voltage data for a P N P N diode shunted by an external capacitor, and an Ovonic diode. The data again compare favorably with the theory. The fact that the experimental oscillations appear as a straight line along the load line (although sometimes they exhibit a slight "bowing") is evidence that our approximation of neglecting the lead inductance is reasonable. If the lead inductance were important the oscillation line would appear as a closed loop. In the Ovonic diode, as the battery voltage is increased above threshold in the relaxation oscillation regime the extrema points on the oscillation line approach each other a). When the extremum point corresponding to minimum voltage becomes too high, however, switching occurs to a voltage near the holding voltage. This behavior is expected for the following reasons. When the current density becomes inhomogenous (filament formation), the conduction current and NDC element voltage mirror this and become time dependent. VD(ic) then exhibits hysteresis during an oscillatory cycle 6). As the battery voltage is increased the threshold voltage decreases and the minimum conduction current reached each cycle increases 6). When it exceeds a "filament sustaining current" [analogue of "domain sustaining field" 1)] the filament is not quenched at the end of the cycle and switching occurs to a voltage near the holding voltage. V(ic)"spirals in" to the switched or ON state6). We find that a spiral approach to the ON state is fundamental to the switching process in general. This behavior is discussed in detail for S-shaped N D C elements in ref. 1, where we make use of an "n subelement" model for inhomogeneous S-shaped N D C elements. Here we model the inhomogenous material by a parallel array of S-shaped conduction current curves. The S-shaped curves have various values of voltage "peak to valley" ratios. In summary, we review the major points and conclusions of our work. First, we have performed a large signal analysis of an arbitrary "S-shaped" N D C element placed in a local environment containing the vital reactive components involved with such an element; the primary circuit. The results provide a firm foundation for the understanding of a wide variety of current instabilities generated in PNPN and Ovonic diodes. To our knowledge, ours is the first detailed large signal circuit analysis applied to an S-shaped N D C element. Our experimental results concur with the approximations we have made. Second, we have found that our theory describes the behavior of both N and S-shaped N D C elements when the important reactive components are identified for each case. For the N-shaped case the primary circuit has the inductor outside the capacitor1), that is, the intrinsic inductance is ignored. The differential equation for the voltage across the N D C element in this case is the same as the equation for the conduction current in the S-shaped case. In other words, we have found that the important circuit parameters in each
CURRENT INSTABILITIES
1003
case p r o d u c e a " d u a l " circuit system where the voltage across the N - s h a p e d N D C element in its p r i m a r y circuit behaves exactly as the current t h r o u g h the S - s h a p e d N D C element in its p r i m a r y circuit. Therefore, our circuit t h e o r y o f a G u n n d i o d e can be t r a n s f o r m e d directly to the case o f an O v o n i c diode, where the high c u r r e n t density filament is the a n a l o g u e o f a high field d o m a i n . W e have benefited f r o m discussions with P. R. S o l o m o n , H. L. G r u b i n , D. A d l e r , E. A. Fagen, a n d H. Fritzsche. W e t h a n k W. Evans for considerable assistance in t a k i n g the data.
References 1) M. P. Shaw, P. R. Solomon and H. L. Grubin, Appl. Phys. Letters 17 (1970) 535; P. R. Solomon, M. P. Shaw and H. L. Grubin, J. Appl. Phys. 43 (1972) 159. 2) This feature was pointed out by G. B. Yntema. 3) M. P. Shaw and I. J. Gastman, Appl. Phys. Letters 19 (1971) 243. 4) Kindly provided by R. F. Shaw at ECD, Inc. 5) See, e.g., K, Homma, Appl. Phys. Letters 18 (1971) 198. 6) M. P. Shaw, H. L. Grubin and I. J. Gastman, IEEE Trans. Electron. Devices, Special issue on amorphous semiconductors, to be published.