Despite being well documented, nonlinearity—its theory and applications—is often viewed as a novelty or strange anomaly. Here, we’ll argue the opposite point of view. Nonlinear and negative resistances may simply be misunderstood, and the standardization of terms and performance claims would further advance their common understanding (see Beyond Ohm’s Law).
This article particularly emphasizes the materials used to fabricate working nonlinear components. The list of materials that may result in non-ohmic behavior is lengthy:
• Simple compounds
• Multi-material composites
• Multilayer structures and junctions
• Germanium (Ge), silicon (Si), and group III-V semiconductors
• Group VI elements (chalcogenides)
• Transition metals
• Organics (carbon-based)
Now, as advances in materials research open the door to new device fabrication techniques, customized performance for the desired application is possible. If silicon does reach its ultimate size limitations at the end of the Moore’s Law curve, nonlinear materials may offer another path to the next generation of electronic devices. In fact, there has been increased observation of nonlinear behavior in the shrinking world of nano-structures and molecular electronics.
COMING TO NONLINEAR TERMS
Negative but linear: The current-voltage (I-V) response of a linear theoretical resistance element may have either positive or negative slope. Furthermore, resistance can be time-invariant or time-variant. For this article, time invariance is assumed.
Linear resistance may have either an ohmic positive I-V slope or a negative slope. A typical, lossy resistor is by definition the ratio of voltage over current (R = E/I, according to Ohm’s Law). Conversely, an op-amp circuit can synthesize a negative resistance through positive feedback. The representation for negative resistance is simply –R. In this case, a linear but negative or decreasing slope appears in the I-V curve, rather than the positive or increasing slope of a lossy resistor.
Interestingly, The Illustrated Dictionary Of Electronics, 7th Edition (McGraw-Hill, 1997) defines “ohmic contact” as one that exhibits none of the properties of a rectifying junction or nonlinear resistance. In other words, ohmic can be defined as the absence of nonlinearity.
“Ohmic region” is defined as the portion of the response curve of a negative-resistance device that exhibits positive (ohmic) resistance. One type of nonlinear current-voltage curve is commonly called a voltage-controlled “N” shaped curve (Fig. 1a). A complementary type of relationship is the current-controlled “S” curve (Fig. 1b). Yet another variation of the S-shape is its mirror image or “Z” curve (Fig. 1c). Subsequent sections of this article will refer to these basic qualitative curves.
The curve in Figure 1a is typical for tunnel diodes. Other familiar forms of negative-resistance devices include the backward diode, Gunn diode, IMPATT diode, and neon lamp. We will not discuss temperature-dependent resistors, or thermistors, which generally remain ohmic at a given temperature.
The schematic symbol for a linear resistor (Fig. 2a) is universally known. Found less often is the symbol for a nonlinear resistor (Fig. 2b), likely because no device manufacturer makes anything called a nonlinear resistor. Figure 2c shows the symbol for tunnel diodes; more about them shortly.
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THE DIODE POWER CONNECTION
A conventional diode is probably the most common type of “nonlinear resistor,” since it has two regions of different I-V slope that constitute its forward- and reverse-bias modes. In practice, piecewise linear approximation is most often used to model nonlinear resistors and diodes. The theory, construction, and applications of conventional diodes are well documented and while nonlinear, they don’t operate in any region of negative I-V slope.
Except for batteries and generators, no power supply would be possible without rectifiers. All power-supply circuits require nonlinearity. At the heart of even a so-called “linear” power supply lies one or more rectifying diodes. Circuits that oscillate or invert dc to ac also require some form of negative resistance. Power conversion at high frequencies is also enabled by nonlinear reactances, including nonlinear capacitors and inductors.
NEGATIVE RESISTANCE DEVICES
The main thrust of this article is to cover devices that are not just nonlinear, but exhibit negative differential resistance (NDR) as well (Fig. 3). NDR is used to describe the downward or negative slope of a section of the I-V relationship. We’re cautious to avoid any suggestion that this behavior constitutes an absolute negative rather than differential negative resistance, with the former more accurately represented by a power source, or simply a negative resistance.
ELECTRONS GO TUNNELING
Leo Esaki’s discovery of quantum mechanical tunneling in germanium (Ge) junctions created the first solid-state device with NDR characteristics.1 The tunnel-diode I-V curve, introduced in Figure 1a, is shown with more detail in Figure 3. As voltage increases across the junction, tunneling current through the junction increases to a peak and then decreases to a characteristic valley.
Tunneling is made possible by the probability of some electrons to exist or “tunnel” under what is otherwise a barrier between valence and conduction bands. The statistical probabilities of quantum mechanics govern where electrons may be at a given time.
Note that the first section of a tunneling I-V curve with positive slope is where the tunneling occurs, while the negative slope region is where tunneling ceases. Thus, it’s really the absence of tunneling that yields the characteristic tunnel diode’s negative differential resistance. S.M. Sze’s Physics Of Semiconductor Devices discusses in more detail three contributing currents—tunneling, excess, and thermal.2
The tunnel diode is a milestone in the history of electronics, resulting in Esaki (along with Brian Josephson) winning the 1973 Nobel Prize in physics. Since the negative I-V slope of tunneling runs counter to Ohm’s law, the initial flurry of interest generated in tunneling devices and applications was appropriate. Still, there are relatively few applications of tunnel diodes.
Early on, the materials inhibited production and, ultimately, the widespread use of tunneling devices. First-generation Ge-based commercial discrete tunnel diodes like the 1N2927 are no longer being made. Over time, advances in other semiconductor materials surpassed that of germanium, and ICs largely replaced discrete semiconductors. Also, difficulties in manufacturing classic tunnel structures have struggled to catch up with dominant CMOS processes.
Since the original Esaki diode, various materials have been used to make tunnel diodes, such as gallium antimony, silicon germanium, gallium arsenide (GaAs) and other III-V materials, and silicon carbide. Furthermore, CMOS-compatible processes have been used to fabricate resonant tunneling devices. Variations on the basic p-n junction tunnel diode that employ double-barrier quantum-well structures include resonant tunneling diodes (RTDs) and resonant interband tunneling diodes (RITDs).
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One important characteristic parameter of NDR is the relative difference between maximum current prior to tunneling and minimum current before tunneling ceases again. This is commonly defined as the peak-to-valley current ratio (PVR) and may be used independently of the underlying mechanism or shape of the I-V curve.
PVR is a common measurement of the goodness of a NDR device. For example, a structure fabricated from III-V materials was reported to have a PVR of 144 at room temperature.3 (Research aside, non-cryogenic temperature operation is a must for practical circuit applications.)
In a further extension of tunneling behavior, triple-barrier devices with multiple NDR characteristics have been created, suggesting applications in logic applications with capability beyond simple binary states. Applications in digital circuits have been envisioned in addition to logic, indicating continued interest in the tunnel-diode concept.
When reducing the doping of a tunnel diode, a diode with a device having larger current in the reverse rather than forward direction results at low bias. Its I-V curve may include a region of NDR with a lower peak-to-valley ratio than a tunnel diode. In other variants, the region of NDR may be absent altogether.
Much like the efforts to improve the manufacturability of Ge-based tunnel diodes, backward (sometimes called “back”) diodes have been fabricated in III-V materials. Backward diodes show that steepness of the NDR region can be modified as desired, including absence of NDR altogether.
An IEEE standard is available for guidance on symbols and test methods for tunnel and backward diodes.4 Certain kinds of test procedures are provided to avoid oscillation, hinting at a major application of NDR devices. The standard further suggests improvements in common ways of reporting NDR, including peak and valley current and voltage that should be specified when reporting on the behavior of these devices.
THE GUNN DIODE
A second class of NDR diodes is based on the transferred electron effect. These devices don’t rely on semiconductor junctions, but rather employ bulk materials with valleys in a satellite conduction band having lower electron mobility than in the main valley. The first such device was reported by J.B. Gunn.5 When the external field increases to a point where lower valley electrons transfer into the satellite valley, reduction in their mobility results in a negative differential resistance.
Unlike the tunnel diode that originally was based on germanium, Gunn diodes have always been based on III-V semiconductors. Gunn diodes based on gallium arsenide and gallium nitride are available commercially for use as microwave oscillators up to the terahertz range.
As early as 1954, another Nobel laureate, William B. Shockley, considered the possibility of negative resistance in semiconductors. Rather than use either electron tunneling or transfer characteristics to produce a diode with NDR, this third type of NDR behavior modifies the transit time within the device to make the current lag the voltage.
The prime example of this behavior exists in the impact ionization avalanche transit-time (IMPATT) diode. Similar to the development of the tunnel diode, many variations on the basic IMPATT structure have been reported. Table 1 lists some examples of these variations within this device class.
Investigation of hot electron effects resulted in a fourth class of NDR devices based on what is called “real space transfer.” In these devices, for example, GaAs-AlGa (aluminum gallium) types of structures, junctions are doped to cause heating and thermionic emission across the device. This class of NDR diodes is capable of S-shaped I-V behavior.
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Subsequent work led to related three-terminal variations, including the negative resistance field-effect transistor (NERFET). The NERFET recorded one of the highest ever peak-to-valley current ratios—up to 7000 at room temperature, and more than a million at a reduced temperature of 77K. Variation in the type of doping further improved the PVR from 7000 to 156,000 at room temperature. Although their widespread commercial application has yet to be fully achieved, these real-space-transfer devices are most promising for microwave applications.
NDR IN SIMPLE JUNCTIONS
While research continues on advanced high-frequency devices, one might also consider the poor man’s kind of tunnel diode named the Negistor. Curve tracing of typical npn transistors, including the 2N2218, 2N2222, and 2N697, shows that the emitter-collector currents fold back to produce NDR behavior.
This finding made it possible to build simple NDR oscillators with very few components (see “The Mysterious ‘Negistor,’” Popular Electronics, Dec. 1975, reprint available at www.keelynet.com/zpe/negistor.htm). These characteristics don’t appear to be supported by any manufacturer’s datasheet, but anyone with an experimental bent may find them interesting to explore.
One can even prepare homemade NDR diodes from simple junctions based on common everyday materials like galvanized sheet metal and use them to produce oscillations in the audio to low-RF frequency range (jlnlabs.online.fr/cnr/negosc.htm). In this case, NDR behavior is explained by the content of transition metal oxides zinc and iron that can form crude tunnel-diode junctions without requiring any complex or sophisticated semiconductor fabrication techniques.
A similar result can be achieved with two JFETs, such as the 2N3819 and 2N3820, connected to make a lambda diode (home.earthlink.net/~lenyr/zincosc.htm). Or, use two PNPN programmable unijunction transistors such as the 2N6027 and 2N6028 (www.sm0vpo.com/_visitors/blocks/lambda_diode.htm). Lambda diodes have also been realized in a combined bipolar and MOSFET circuit (www.du.edu/~etuttle/electron/elect10.htm).
While these transistors yield typical N-shaped I-V curves, developments in heterojunction-emitter bipolar transistors (HEBTs) have yielded S-shaped NDR behavior. Some of these devices feature multiple regions of NDR that invoke applications in multi-state logic.
In devices like varistors, extremely nonlinear I-V properties of metal oxides are commonly used to protect against unwanted transients. Commercially available devices, made in either bulk or multilayer form, come in standard packages. Typical transition metal-oxide varistors are based on zinc with additions of bismuth, manganese, and cobalt.
Varistors can be considered multi-junction devices. Each of the numerous grain boundaries in the microstructure contributes to both series and parallel connections. Besides being used as passive protection devices, the recent development of pulse-induced switching in these materials will lead to the development of new types of memories. Although conventional varistors are nonlinear, they’re generally not capable of NDR behavior.
VAPOR PHASE AND PLASMA DEVICES
NDR isn’t limited to solid-state devices. Unlike the N-shaped I-V behavior of tunnel devices, gas tubes may exhibit S- or Z-shaped behavior. The mechanism within plasma devices is a complex space charge configuration. Gas discharge tubes are useful for high-voltage protection, including shunting transients induced by lighting to ground. Even conventional neon lamps show NDR behavior, which has been exploited to make simple oscillators.
By now it should be clear that although they’re considered “passive” devices, two-terminal components can nonetheless offer very dynamic properties. Table 2 summarizes the types of negative-resistance devices discussed in this article, including their capability for NDR behavior, the typical shape of the I-V curve, and comments on construction and applications.
Clearly there is an opportunity to improve on the standardization of terms and parameters associated with NDR functionality at the device level so they can be more accurately compared to each other. Though some papers have offered a fundamental understanding of NDR, it’s unlikely that any single unified theory will ever be realized due to the vast array of behaviors, materials, and mechanisms involved.
Roger L. Franz, a former engineering manager in Motorola’s Mobile Device Business, holds a BA from Grinnell College, Grinnell, Iowa, and an MS from Northwestern University, Evanston, Ill.
1. L. Esaki, “New phenomenon in narrow germanium p+n+ junctions,” Physical Review, vol. 109, pp. 603-604, 1958.
2. S.M. Sze, Physics of Semiconductor Devices, 2nd ed., New York: John Wiley & Sons, 1981.
3. H.H. Tsai, Y.K. Su, H.H. Lin, R.L. Wang and T.L. Lee, “P-N Double Quantum Well Resonant Interband Tunneling Diode with Peak-to-Valley Current Ratio of 144 at Room Temperature,” IEEE Electron Device Letters, vol. 15, pp 357-359, Sep. 1994.
4. “IEEE Standard on Definitions, Symbols, and Methods of Test for Semiconductor Tunnel (Esaki) Diodes and Backward Diodes,” IEEE Transactions on Electron Devices, vol. 12, pp. 373-386, June 1965.
5. J. Gunn, “Microwave Oscillations of Current in III-V-Semiconductors,” Solid-State Communications, Vol. 1, p. 88-91, 1963.