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This article has been published in Nanotechnology (1996) 7 pages 325-339, and is also available in HTML at: http://www.zyvex.com/nanotech/helical/helical.html. The HTML version differs in some respects from the published version.
Copyright 1992 by Xerox Corporation and the Institute for
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Conventional devices actually perform more poorly. Even an idealized device which used a one volt power supply and dissipatively discharged a single electron to ground during a switching operation would dissipate one electron volt per switching operation. At T=300 Kelvins, this is 40 × kT per switching operation or about 160,000,000 watts for a computer with 1017 logic elements operating at 10 gigahertz. If each switching operation involves hundreds of electrons then energy dissipation enters the multigigawatt range.
New thermodynamically reversible circuits (including CMOS, nMOS and CCD-based logic circuits(Hall 1992, Merkle 1993c, Younis and Knight 1993, Koller and Athas 1992, Merkle 1992)) would fare better, but these circuits still have dissipative losses caused by the resistance of the circuit. While resistance in sufficiently small wires can be very low(Sakaki 1980), if such wires are connected to each other, to logic elements or to larger structures it is common to find resistances of the order of 13K (half of h / e2, where h is Planck's constant) (note that no claim is made that the successful operation of such circuits must fundamentally require resistances of this magnitude, we simply note that shrinking current circuits to a small scale would result in such resistances: further research in this area might be successful in dealing with this problem). Assuming that 100 electrons were required to charge and discharge the wires and capacitive loads in each logic element, and assuming a resistance of approximately 13K, we would still find our 1017 gate computer dissipating tens of megawatts even using these particular thermodynamically reversible methods.
If the exponential trends of recent decades continue, energy dissipation per logic operation will reach kT (for T=300 Kelvins) early in the next century(Landauer 1988). Either energy dissipation per logic operation will be reduced significantly below 3 × 10-21 joules, or we will fail to achieve computers that simultaneously combine high packing densities with gigahertz or higher speeds of operation. There are only two ways that energy dissipation can be reduced below 3 × 10-21 joules: by operating at temperatures below room temperature (thus reducing kT), or by using thermodynamically reversible logic. Low temperature operation doesn't actually reduce total energy dissipation, it just shifts it from computation to refrigeration (Halliday and Resnick 1988). Thermodynamically reversible logic elements, in contrast, can reduce total energy dissipation per logic operation to << kT. This paper analyzes a proposed thermodynamically reversible single electron logic system. To achieve high reliability while switching single electrons, we analyze operation at ~1 Kelvin.
A wide variety of reversible device proposals have been made(Drexler 1988, Likharev 1982, Likharev et al. 1985, Drexler 1992, Fredkin and Toffoli 1978, Seitz et al. 1985, Merkle 1993c, Younis and Knight 1993, Koller and Athas 1992, Merkle 1992). The present proposal is abstractly similar to billiard ball logic(Bennett 1985, Fredkin and Toffoli 1982), and in particular uses the concept of a "switch gate" or "interaction gate."
In billiard ball logic(Fredkin and Toffoli 1982), a set of billiard balls are fired into a set of immovable reflectors at a fixed speed. As the billiard balls bounce off each other and off the reflectors, they perform a reversible computation. Provided that the collisions between the billiard balls and between the billiard balls and the reflectors are perfectly elastic, the computation can proceed at a fixed finite speed with no energy loss.
As originally proposed, this "ballistic" model of computation suffers from the shortcoming that the positions of the reflectors and the initial velocity of the billiard balls must be perfectly accurate. In the real world, the computation would rapidly deteriorate into chaos unless some restoring force maintained the alignment of the billiard balls. Landauer(Landauer 1981) proposed exactly this, suggesting that the billiard balls should remain in the trough of a moving periodic potential (illustrated in Figure 4).
Billiard ball logic is usually thought of in connection with Fredkin gates(Fredkin and Toffoli 1982). Fredkin gates are conservative three-input three-output gates that are logically complete, i.e., any computation can be implemented by an appropriately connected set of Fredkin gates. An appropriately arranged set of reflectors can be used to implement a Fredkin gate from billiard balls. Rather than considering the Fredkin gate we will focus instead on the switch gate(Fredkin and Toffoli 1982). This can be used to make a Fredkin gate (and so is also logically universal) but it is physically simpler and easier to implement.
A Fredkin gate is illustrated in Figure 1, a switch gate is illustrated in Figure 2, a method of implementing a Fredkin gate from switch gates is shown in figure 3, while a billiard ball being carried along by a periodic potential is shown in Figure 4.
While the abstract concept of billiard ball logic has been known for some time, the idea of implmenting this approach by replacing the billiard balls with charged particles is surprisingly recent(Merkle 1993c) (though Landauer(Landauer 1981) did refer to an unspecified short range repulsion). In the present proposal, we replace the billiard ball with a single electron. While other charged particles or charged packets of particles can be used, we will focus on the single electron implementation because of the obvious long term performance advantages. The concept that future electronic devices might use single electrons is becoming more accepted(Likharev and Claeson 1992, Washburn 1992, Grabert and Devoret 1992). This is the first description of a thermodynamically reversible switching device using single electrons. Not only are we switching single electrons, we are also using single electrons to control the switching (unlike many proposals which require many electrons to switch a single electron). It is plausible that thermodynamically reversible single electron switching devices will be the ultimate evolutionary end point of electronic logic devices. The authors emphasize that other proposals for single electron thermodynamically reversible logic devices will certainly be advanced, so the present proposal must be viewed as the first in a class rather than "the" one and only design for a single electron thermodynamically reversible logic device.
While the analysis here focuses on the case of a "charge packet" consisting of a single electron, the use of a multi- electron charge packet would clearly be simpler in terms of a more near term implementation. In particular, charge packets are commonly used in CCD's. By making a series of design choices in favor of near term feasibility we can arrive at a design which is basically a thermodynamically reversible type of CCD logic. This possibility is discussed more fully in (Merkle 1993c). The performance of such a system would likely fall well short of ultimate limits for pragmatic reasons. The present paper focuses on an approach which should be feasible in the long term (though less likely to be practical in the near term) and which offers the possibility of greatly superior performance and, in particular, the possibility of very low energy dissipation while still operating in the gigahertz range or higher.
The picture of clock distribution being proposed here is new (though it is abstractly similar to the rotating magnetic field used to clock and power magnetic bubble devices(Bobeck and Scovil 1971, Kinoshita et al. 1976)). The computing element is immersed in a rotating electric field. The electric field, and hence the clock, is available at every point in the device and yet no wires are required to distribute it. This reduces energy dissipation and eliminates the space that would otherwise be occupied by the clock distribution network.
A rotating electric field is rather different from a periodic moving potential. In order to achieve the same effect (i.e., to move electrons forwards along their path) we propose the use of helical paths, with the electric field rotating at right angles to the axis of the path. The electron will be confined to part of a single turn of the helix and as the electric field rotates the electron will be moved along the helical path much as water is moved along an Archimedes screw. This is illustrated in figure 5.
If we were to examine the potential at any point along the helical path we would find that it varied in a sinusoidal fashion much as the potential varies along a straight path under the influence of a moving periodic potential. The use of a periodic moving potential with straight paths (as shown in figure 4) and the use of a simple rotating electric field with helical paths produce very similar results. In either case, the electron is moved along the path.
There are other methods of providing a moving periodic potential that do not involve distribution of clock signals through wires. A simple proposal would be to have a circuit on the surface of a rotating disk, with an opposing disk with fixed charges on it. The relative motion of the two disks would result in a moving potential that would clock the circuit. Alternatively, instead of disks two tubes of differing diameter, one tube placed inside the other, could be used. Circuitry could be placed on the inner surface of the outer tube, while fixed charges could be placed on the outer surface of the inner tube. Again, relative motion of the inner tube with respect to the outer tube could provide a moving potential. This approach also has the advantage that the tubes could be reduced in radius to a very small size (nanometers) and the rotational speed of the inner tube could be made very high (gigahertz) without any fundamental problems. Many such tubes could be stacked adjacent to each other, and charged particles being swept along a potential on one tube could move to an adjacent tube, provided that the movements of the clocking potentials in the two adjacent tubes were appropriately synchronized. A different approach would be to operate the circuit on a piezomechanical surface in which surface acoustic waves created a moving electric potential.
We will not consider the many possible alternatives in this paper.
The reverse of this operation, in which three incoming helical paths produce two outgoing helical paths, is similar but "run backwards" in time. Because the basic operation of a switch gate is reversible, it can be operated in either the forwards or reverse direction. We will therefore not explicitly analyze this process. Note that in reverse operation it is essential that the presence or absence of a charge carrier on the data path be correctly correlated with the path (left or right) along which the switched charge is entering the switch gate. This constraint can be met (as illustrated in figure 3 showing the design of the Fredkin gate).
It is worth noting that irreversible operations are sometimes convenient. A basic irreversible operation in helical logic merges two incoming helical paths into one outgoing helical path (and dispenses with the data path). Such an irreversible logic operation, if properly designed, should have an energy dissipation which approaches the fundamental limit: ln 2 kT. Helical logic has the valuable ability to degrade gracefully when irreversible operations are required.
We will focus primarily on fundamental issues of device performance while neglecting the issues involved in manufacturing any specific device. Thus, we will simply assume that the ability to economically manufacture atomically precise semiconductor material, with dopant atoms placed at atomically precise lattice coordinates, is available. Such a manufacturing technology is not available today but should be available at some point in the future(Drexler 1992, Merkle 1994). In the long run, if we are to achieve the maximum performance possible from semiconductor devices, we will have to develop and use some sort of molecular manufacturing technology. This is true almost regardless of the specific details of the device proposal. The more precisely a device can be fabricated, the better the achievable performance. The limit of this trend will be devices in which each atom is in the right place.
There are many choices for the materials that comprise the helical path and the surrounding medium. An obvious choice is GaAs and AlAs(Sze 1990). Electrons prefer GaAs to AlAs by about 0.3 ev(Van de Walle 1989), so by operating at a sufficiently low temperature complete confinement of the electrons could be achieved. Another choice would be Ge and Si(Sze 1990, Bean 1992). Other possibilities that should become feasible in the future with the advent of molecular manufacturing(Drexler 1992, Feynman 1960), would be Si and GaP(Sze 1990), Si and ZnS(Sze 1990), or even C (in the form of diamond) and vacuum. The latter would offer significant advantages because of the low dielectric constant of vaccuum, although the negative electron affinity of the hydrogenated diamond (111) surface suggests that either the use of holes as the charge carriers or the modification of the diamond surface to achieve a positive electron affinity (i.e., fluorinated diamond (111)) would be advantageous. If the helical channels are diamond with hydrogenated diamond (111) surfaces, an electron in the channel would tend to escape into the surrounding vacuum. The idea that a hole might perform a similiar feat is less plausible. Diamond channels with fluorinated surfaces in vaccuum using either electrons or holes (or both) as charge carriers might be an attractive choice once our manufacturing technology is able to build the required structures.
The negative electron affinity of the hydrogenated (111) diamond surface also suggests the use of an evacuated channel surrounded by walls of diamond. This approach means the channels are holes bored in a block of diamond. The motion of electrons through evacuated channels might prove advantageous by reducing electron/channel interactions. The creation of a helical tube with relatively smooth walls in a block of unstrained diamond would require the use of more than just the (111) plane. Other surfaces (e.g., (100) or (110)) could be used and would permit a tube with smoother walls. Approaches using strained diamond (e.g., diamond which is curved by, for example, the introduction of appropriately placed dislocations (Merkle 1993a)) would be another alternative.
The use of other charge carriers produces a wider range of possibilities. For example, the charge carrier could in principle be a single proton. The channels would then be pores through which H+ could easily move. The increased mass of H+ (as compared with an electron) would simplify confinement by decreasing the effective distance through which the charge carrier could tunnel. Channel size and interchannel distances could both be made much smaller; proton-proton repulsive interactions would therefore be larger and operating temperature could be increased. The smaller size and higher operating temperature of a helical logic system based on hydrogen ions would compensate to some extent for the slower speed, and the result might be advantageous in some applications. Many other small ions could also be used.
The use of charge packets made up of many electrons (or holes) should provide a method of implementing helical logic using today's technology at higher temperatures. The use of charge packets is common in CCDs. It is an open question whether today's technology can economically mass produce the complex structures needed for helical logic (though research devices should be feasible). However, a 2D version of helical logic which used spiral paths should be feasible with today's technology. The implementation of a planar switch gate should likewise be feasible. More research on this point would be worthwhile.
Whatever the materials choice and manufacturing technology, the primary requirement is that a charge carrier be confined to a helical path, and be able to interact with other charge carriers in other helical paths.
It is important to distinguish between charge transport in helical logic and charge transport along a wire, even a very fine wire. The mobility of an electron confined to a one-dimensional wire can be much higher than in the bulk material(Sakaki 1980). However, even in this case the electron is confined in only two dimensions and can move freely along the third. In helical logic, by contrast, the electron is confined in all three dimensions. Viewed from the frame of reference of the moving electron, it is simply sitting at the bottom of a potential well.
Lattice vibrations induced by the moving charge will cause transport losses. Treating the electron as an isolated charge subject to an oscillating force F, we have(Drexler 1992, page 164):
Prad is the radiated power, F is the force applied to the charge, is the frequency in radians per second, is the density, and M is a modulus of elasticity.
For an electric field E of 108 volts/meter (106 volts/centimeter, well below the breakdown strength of diamond) we have a force F on a single electron of 1.6 × 10-19 × 108 = 1.6 × 10-11 N. The density of diamond is about 3,500 kg/m3. The frequency in radians per second for a 100 picosecond time to make a single rotation is 2 × 1010. M is about 1012 Pascals. Substituting these values into equation (1) yields a radiated power of roughly 2.4 × 10-18 watts, or 2.4 × 10-28 joules per charge carrier per rotation of the electric field.
This treats the electron as an isolated point charge, and also uses an approximation which should be accurate at frequencies well below the Debye frequency. The frequency selected is below the Debye frequency for diamond, while the error caused by the point charge assumption is conservative (in the sense that this assumption makes the energy dissipation higher than one would otherwise expect).
Lower power dissipation should be feasible. We could maintain approximate charge neutrality by placing a fixed charge in the neighborhood of the electron. A donor atom contributing a positive charge will effectively cancel the negative charge of the electron. This is likely to produce a lower energy dissipation, as acoustic energy radiated from a dipole is lower than that radiated from a monopole (when the wavelength of the radiation is longer than the distance between the charges in the dipole). Alternatively, we could represent a logic "1" by the presence of an electron and a hole. The resulting dipole could then be transported along two helices ("double helix logic") or both the electron and the hole could be confined in a single helix, one of them being half a turn ahead of the other. In the latter case, two types of dipole would be possible: one in which the electron precedes the hole, and another in which the hole precedes the electron. While it might appear paradoxical to have a helix which can simultanesouly confine both an electron and a hole, diamond helices with fluorinated surfaces in vacuum should (as noted earlier) be able to do exactly that.
A frequency of 1010 hertz, along with the assumption that a single turn of the helix has a diameter of about 100 nanometers, implies that the electron is moving at a speed of about 3-4,000 meters per second (below the speed of sound in diamond). The dissipation mechanisms that might come into play for significantly higher speeds are not analyzed here.
Accelerations caused by variations in the helical structure for the purpose of altering the direction of charge transport will result in accelerations of the charged particle that might well result in radiative losses. Radiative losses are discussed further under switching losses, as accelerations of the charged particles during switching can likewise result in radiative losses.
Dielectric losses for bulk silicon and diamond are described in (Braginsky 1985, Braginsky 1987, Gurevich 1979, Gurevich and Tagantsev 1991) by the equation:
Where is a dimensionless anharmonicity parameter with typical values between 10 and 100, is the frequency in radians per second, e is the dielectric permittivity of the crystal, the density, h is Planck's constant, v the speed of sound, k is Boltzmann's constant, T the temperature in Kelvins, and TD the Debye temperature. Note that Q ~ 1/(tan ).
In the case of diamond and approximating the value of as 100(Braginsky 1987) we can approximate the loss factor. If we use = 2 × 1010, k = 1.38 × 10-23, T = 1 K, e = 5.7, = 3,510 kg/m3, h = 6 × 10-34, TD = 2340, v ~ 15,000 m/s; then we get tan ~ 10-20. (Experimental tan 's for saphire of less than 10-9 at a frequency of about 10 gigahertz at a temperature of 1.5 Kelvins have been observed(Braginsky 1987). Saphire has a fundamentally higher tan because of its crystal symmetry). If we assume (rather generously) that each switch is a cube 100 nanometers on a side, and if we further assume that ten times this volume might be required for interconnections between switches (again rather generous), we have a total volume per switch of 107 cubic nanometers. The energy stored by an electricic field of 108 volts/meter in this volume results in an energy of about 2.5 × 10-15 joules per gate (assuming a dielectric of 5.7, as before). Multiplied by our dielectric loss factor, we get an energy loss of under 10-34 joules per cycle per gate.
The form of the equation governing the dielectric loss varies greatly depending on the crystal symmetry. For example, centrosymmetric crystals of the D6h symmetry group have a dielectric loss that falls off as T9 (linear falloff in frequency, but ninth power in the temperature), while dielectric loss for crystals of the Ci symmetry group falls off as 5T(Gurevich and Tagantsev 1991).
The presence of the helical paths creates a non-uniform structure, and hence induces additional losses. A loss mechanism suggested by Soreff(Soreff 1994) is caused by forces acting between induced dipoles. Even when charge carriers are completely absent the rotating electric field will induce dipoles in the helical paths because the dielectric constant of the path and of the surrounding medium typically differ. A single half-turn of a helical path of the dimensions considered here can be approximated as a column about 100 nm long by 10 nm deep and 10 nm wide. If the electric field is 108 v/m and we assume the disparity of the dielectric constants of the path and of the medium is approximately 5.7 (as would be the case for diamond helical paths), such a volume would have an induced dipole moment of roughly 5 × 10-26 C-m. Two such dipoles at a distance of roughly 100 nm (i.e., the opposing halves of a single helical turn) will experience a repulsive force of under 10-12 N (using a formula derived from that for the interaction energy given in (Israelachvili 1992)). Recall that the helical track is already being subjected to forces caused by the charge carrier, estimated above as 1.6 × 10-11 N, and that the energy dissipation is a function of the square of the force. Thus, the inter-track repulsive forces caused by induced dipoles are in this approximation smaller than the forces caused by a charge carrier by at least an order of magnitude. Because the inter-track spacing is small compared with the acoustic wavelength, dissipation from this mechanism should be further substantially reduced.
The actual dielectric losses in a complex three dimensional structure consisting of numerous very small logic gates are certain to be different from the value estimated here for a perfect diamond crystal. However, it seems unlikely that such losses must necessarily be much larger and they can likely be made much smaller. The wide variation of dielectric loss among crystal types suggests that careful selection and design of materials can be used to make systems with very low dielectric losses. The uncertainty in the dielectric loss remains substantial, however, and the actual loss could potentially be significantly higher than estimated here.
The reader who wants to think about a single specific geometry while reading the analysis in this section can read the geometry of a specific switch gate in the first three paragraphs of the section titled "Quantum Mechanical Analysis of the Operation of a Specific Switch Gate."
Switching losses occur because the two electrons are excited by their interaction and will dissipate energy when they fall back to their ground state. If we can keep the two electrons from entering an excited state, we can avoid energy dissipation.
A rather interesting observation is that switching losses caused by this mechanism can in principle be avoided almost completely even if the two electrons do enter excited states. An electron is in an "excited" state only with respect to some potential energy function. For any given wave function, however, there exists a potential energy function for which that wave function is the ground state. Thus, if we knew the wave function of each electron as it left the switch gate, and if we could engineer the path along which the electron moves to have the appropriate potential energy function, then the energy dissipated by the electron could in principle be eliminated.
Put more generally, if we know the state of a system then we need not (in principle) dissipate any energy at all.
Unfortunately, it would seem that the wave function of an electron leaving a switch gate depends on the presence or absence of the other electron. Even if both electrons entered the switch gate in their ground state, and even though the evolution of the wave function is deterministic, if we look only at one of the departing electrons we will find it in one of two states depending on the presence or absence of the other electron.
This can, in general, be solved by the use of the "interaction gate"(Fredkin and Toffoli 1982) illustrated in figure 7.
In the interaction gate two electrons enter along two paths (A and B), and depart by four paths. Again, the interaction gate can be used to implement a Fredkin gate, and so it is logically complete. What makes the interaction gate interesting is the following property: if we know that an electron is present on a particular output path, then we know the inputs to the gate. For example, if an electron leaves the interaction gate along the A left path, then we know that there was an electron on path B (or else the electron would have left the gate on the A right path). Thus, we can engineer the precise geometry and potential of the A left path knowing exactly what state the electron is in when it travels along that path.
Similar observations hold for all other ways in which an electron can leave the interaction gate, and so by appropriate engineering of the outgoing paths we can essentially eliminate energy dissipation from this source.
When a charge is switched down one of two alternate paths (as happens to the data electron in figure 6) it is subjected to an arguably unpredictable acceleration (we don't know which path it's going to take) which will in turn generate an arguably unpredictable pattern of radiation. If we assume that the accelerations involved are similar to those that would occur in an oscillating dipole, we can readily estimate the magnitude of this loss by using the formula for the power radiated by an oscillating point charge(Feynman 1963), page 32-3]:
Dividing the radiated power by the frequency gives the energy dissipated per cycle, which is:
For f = 10 gigahertz and r = 50 nanometers (and neglecting the dielectric constant of the medium) this results in losses of about 10-35 joules per cycle.
Although this is a reasonable approximation to the energy losses caused by the acceleration of a single charge during the switching process, as noted earlier the switching frequency and the frequency of the externally applied electric field need not be exactly the same. Likewise, the distance which the switched and switching charges must traverse during a switching operation is different from the radius of the helix, and the acceleration profiles need not be sinusoidal. Further, modifications to the helical structure to transport charge carriers to various more or less random locations required by the computation will result in accelerations unrelated to switching.
Despite these admittedly rough approximations, the low dissipation computed suggests that this mechanism will not be the dominate source of energy dissipation under the conditions considered in this paper.
Because the energy dissipated per cycle is a function of the square of the charge, and because it might happen that many switches simultaneously switch charge in the same direction (thus effectively increasing the charge that is being accelerated) the energy dissipation from this mechanism might be increased significantly. This could be reduced by careful design of the switching operations to reduce the likelihood of such an event. Even stronger, if double helical logic is employed then a single switching operation could be so designed that it simultaneously accelerated both an electron and a hole in the same direction and by the same amount. This should effectively reduce energy dissipation from this mechanism by many orders of magnitude.
We assume that acoustic losses caused by the switching operation are similar to or smaller in magnitude than the acoustic losses caused by charge transport.
This geometry can be visualized in the following way: think of the condition paths as being wrapped helically around one tube, and the data paths as being wrapped helically around a second tube of slightly larger diameter. Now, insert the smaller tube into the larger tube and align the switching region of the data paths with the appropriate region of the condition paths. As the diameters of the two tubes are different, the electrons moving along a helical path on one tube can't move onto a helical path along the other tube. Only their electric fields will interact, allowing an electron on one tube to "push" the electron on the other tube down one of two alternative pathways.
Another way of thinking of the geometry of this switch gate starts by considering the diagram of figure 6. Move the two condition paths above the plane of the paper (move the two condition paths towards you as you look at figure 6 -- this movement is orthogonal to the plane of the paper) by some short distance. This effectively moves the condition paths out of the plane of the data paths, and allows the condition and data paths to move up and down freely without any risk that they will collide and merge their contents. Note that "up" and "down" in this context are defined relative to figure 6 -- the switched data electron moving along the left path is moving "up". Finally, wrap figure 6 into a helical tube by raising the upper left and lower right corners towards each other.
The analysis that follows shows that a "plausible" potential exists which simultaneously provides a low energy dissipation and a low error rate at a reasonable switching speed. The underlying heuristic guiding its design was to minimize excitation of the switched electron during the switching process. The switched electron is assumed to have a significant probability of being excited, a probability which must be analyzed and minimized. Such excitation leads to both energy dissipation and to errors. By minimizing the excitation, both problems can be reduced.
The authors expect that significantly better potentials are feasible.
Standard SI units are used throughout.
The switching time proposed is a few orders of magnitude slower than the period of a single oscillation of the switched electron when excited to the first excited state. This speed allows us to use the adiabatic approximation in computing the wave function of the switched electron, in estimating the probability of error, and in estimating the energy dissipation. Faster switching times should be feasible but would require a more detailed (and complex) analysis.
Variations in potential along the z axis are limited in frequency by the spacing of the atoms, and are further effectively limited by the fact the wave function will have some spread along the z axis. As a consequence, we have limited ourselves to a potential that varies relatively slowly with changes in the z coordinate. Variations in potential along the x axis involve tens of nanometers, and so could reasonably be engineered by appropriate placement of individual atoms. The potential along the y axis is assumed to confine the electron to a fixed distance from the axis of the helix, i.e., the electron is confined to a helical ribbon.
where Pn is the probability of ending in state n (assuming the initial state is m), Cnm is the matrix element, E is the energy of a state, H is the system Hamiltonian, and vn and vm are wave functions. This gives an upper bound on the probability when the rate of change in the Hamiltonian (H/t) is a constant over some interval and zero before and after. Transition probabilities for systems in which H/t increases and decreases smoothly over a time t >> /E) are much lower; probabilities can also be larger or smaller after a series of abrupt changes, as phases add or cancel. For the present analysis, the maximum value of Pn encountered in a series of sample times, Pn,max, is used to estimate the probability of occupancy of that state at the end of that time interval. A more detailed analysis preserving phase information would be of interest, but should for the present system yield results of the same order as this estimate.
The following analysis describes the Hamiltonian and associated switched-charge wave functions along a one-dimensional coordinate. Wave functions are computed in the adiabatic approximation, neglecting the perturbing potential imposed by acceleration and charge mass (this imposed potential has a magnitude less than 5 × 10-23 J, significantly smaller than the potentials imposed by either the switching charge or the material of the switch). Changes in this perturbing potential, however, provide a significant component of H/t and hence substantially affect excitation probabilities.
We will call the potential created by the switch gate acting on the switched charge in the absence of the switching charge the "underlying potential." This potential is created by appropriate design of the switch gate. Because the switch gate is bilaterally symmetric, the underlying potential must necessarily be bilaterally symmetric as well.
Ideally, the switching charge would steer the switched charge down the chosen path without changing the shape of the potential well. From the frame of reference of the switched electron, almost nothing would have happened and excitation would be minimal. If the underlying potential were harmonic, and the potential imposed by the switching charge had a linear gradient, then the approach of the switching charge would have exactly this effect (a linear gradient applied to a harmonic well results in a new harmonic well of the same width but moved laterally: exactly what we desire).
In practice, the gradient created by a point charge is nonlinear. To some extent, this can be compensated by adjusting the underlying potential so that the resulting total potential (the underlying potential plus the potential created by the switching electron) is harmonic. This cannot be done completely, for the underlying potential is bilaterally symmetric and the wave function of the switched electron will have a significant component on both sides of the center of the switch gate (at least during the earlier phases of switching). However, the region of the underlying potential in the vicinity of the switched electron and on the same side of the centerline can be made exactly harmonic by appropriate modifications to the underlying potential. Second order terms can be cancelled by appropriate changes in the underlying potential, but higher-order terms cannot be cancelled during the early stages of switching, owing to the symmetry constraint on the underlying potential. (Effective cancellation of third-order terms becomes feasible when the switched charge has shifted by more than the characteristic radius of its ground-state wave function.) The residual sources of H/t include third-order terms (during the early stages of switching), together with changes in the acceleration-induced potential and changes in the potential associated with the growth of a barrier as the incoming well in the underlying potential splits to form two outgoing wells.
The potential combines several distinct components. The first is the electrostatic potential imposed by the switching charge:
We assume e is 5.7 (corresponding to the dielectric constant of diamond) and r is 4 nm (a gap in the y direction sufficient to reduce cross-channel tunnelling to low values). The parameter z is a dimensionless measure of distance along the path, entering into the dynamical analysis through a choice of dz/dt.
All other components are part of the underlying potential and are constrained to be even functions about the midpoint. These include a harmonic term:
and a recursively-defined correction term:
g(z) = - ksxswitched (z) and x' = 2xswitched(z) - |x|
This term cancels the difference between the electrostatic potential and a linear gradient with a slope of g J/m in the region between x = 0 and x = 2xswitched(z), yielding a total potential that is exactly harmonic within these bounds. There does not appear to be a unique "right" choice for the value of the total potential outside these bounds. We have (somewhat arbitrarily) chosen the underlying potential so that [Vtotal(xswitched(z)+offset, z) - Vtotal(xswitched(z), z)] + [Vtotal(xswitched(z)-offset, z) - Vtotal(xswitched(z), z)] = Vharm(offset) + Vharm(-offset).
Finally, the walls of the potential well are steepened in regions where the ground state wave function has a small amplitude by adding
This term substantially raises the energies of the higher excited states without greatly affecting the ground state.
These various potentials have more or less randomly altered the potential at the bottom of the harmonic potential well, i.e., Vtotal(xswitched (z), z) might vary as z varies. We desire, however, that the potential at the bottom of the well remain constant. To this end, we define Vzero(z) as having that value required to make Vtotal(xswitched (z), z) equal 0. Note that Vzero(z) is a function of z only and does not change as x changes.
The overall potential is:
Vtotal(x,z) = Vharm(x) + Velect(x,z) + Vcorr(x,z) + Vwall(x,z) + Vzero(z)
The total potential has continuous first derivatives so long as the gradient g is numerically equal to Velect(x,z)/x evaluated at x = 0. This condition can be maintained during the early evolution of the potential, but must be violated at later times. The difference between these gradients defines the magnitude of a wedge potential (creating a peak at x = 0) that forms an implicit component of Vtotal.
(A larger version of Figure 8 is available).
(A larger version of Figure 9 is available).
(A larger version of Figure 10 is available).
Table 1. Defining points of natural cubic splines (Press et al. 1992) describing the trajectories of switched and switching charges in the model potential (rounded to three digit precision). The parameter z is dimensionless.
In the initial (pre-wedge) phase of the switching interaction (through z = ~0.05), maintaining a zero wedge potential makes xswitching and xswitched functions of one another (a relationship approximated by the given splines). The acceleration profile is chosen to limit the the jerk resulting from the motion of the well minimum. (Note that the "wedge potential" is not an actual potential, but is merely a consequence of combining the potentials already defined. It is useful, however, to give it a name).
In the following (disengagement) phase, the growth of the wedge potential continues the motion of the switched charge away from the midplane, and a substantial and growing region around the well minimum is exactly harmonic. Changes in well shape outside this region cause residual excitation, but these decline as the exactly harmonic region expands into the tails of the ground-state wave function. (In particular, the discontinuity in the potential gradient associated with the peak of the wedge results in a fast rate of change of potential as seen in the well frame.) The disengagement phase ends as the magnitude of the ground-state wave function at x = 0 becomes negligible (when z ~ 0.15).
In the final (separation) phase, excitation is small, and (according to the energy-based error bound described above) errors are minimized by choosing a design that maximizes the quantum number n of the lowest excited state centered in a secondary well. Placement of the switching charge so as to create two secondary wells of equal depths is nearly optimal during this phase; in the model potential, this condition determines xswitching for z > 0.2. The lowest state that can become bound in the wrong well, n = 9.
Another method would be to use fiber optics. This eliminates concerns about putting wires into an intense rotating electric field. Simply by having an input fiber which started well outside the region of the rotating electric field and terminated within the active region it would be possible to selectively create electron-hole pairs in the semiconductor by injecting light into the other end of the fiber. Absorption of photons can be done in relatively small structures; for example, a one micron thickness of GaAs can absorb roughly a third of the incident light(Casey and Panish 1978) page 46]. If the GaAs were part of a helical structure, the generated electrons and holes would then be separated by the electric field and would move to opposite sides of the helix. This method of generating electron-hole pairs will produce packets of irregular size. By making too many electrons and holes and then selectively re-combining unwanted electrons and holes, we could reduce the packet size to the desired value. Re-combination will occur when the diameter of the helix becomes small enough to allow the electrons and holes to recombine. By controlling the diameter of the helix, the number of electrons and holes that remain in the packets can be controlled. An alternative method would be split the helix and allow the packet (through self repulsion) to break apart into smaller pieces. A sufficient number of splits would produce packets that had only a single electron or hole in them. The result would be several helices each one of which contained exactly zero or one charge carrier. This rather error-ridden input could then be converted into a more reliable result using appropriate logic circuitry.
Having once generated a pair of charge carriers, we could simply continue to let the pair be switched throughout further logic operations. Double helical logic would require the use of two helical paths and two switching elements, increasing the volume and component complexity. By using single helical dipole logic (i.e., a single helix carrying both an electron and a hole to represent a "1") the leading charge carrier would be switched through a switch gate and when the electric field rotated through an additional 180 degrees, the second carrier of the pair would be switched in exactly the same fashion.
For output, we need merely recombine electrons with holes next to an optical fiber. The optical fiber can then carry the resulting photons outside the rotating electric field where conventional detectors and electronics can be used. In single helix dipole logic, a simple method of combining electrons and holes would be to terminate a helical path carrying a pair. The leading charge carrier, upon reaching the end of the helical path, would be unable to continue. The trailing carrier would then catch up. When the two met, the electron and hole would recombine and generate a photon.
Both input and output, depending on how they're implemented, might suffer from a relatively high error rate. This can be corrected by using various coding and multiple transmission schemes that provide sufficient redundancy to correct the errors.
Simple, robust I/O using optical methods should be feasible even in the presence of the strong rotating electric field required by helical logic.
The optical methods of input and output described here are highly dissipative. Many entering photons will fail to generate an appropriate electron-hole pair; when a pair is generated in the electric field the resulting current flow will be dissipative; the recombination of electrons with holes to adjust the size of the charge packet is dissipative; and when electrons are combined with holes to generate optical output many of the resulting photons will be lost. While simple, this method of input and output is far from reversible and will impose energy limitations on the I/O bandwidth. It is often feasible in principle for I/O to be done in a reversible fashion. If two reversible processors wish to communicate with each other in a logically reversible manner, then there is no fundamental requirement that the process be dissipative. Further research is needed to provide a good low energy method of input and output.
Three main categories of energy dissipation were considered: transport losses, dielectric losses, and switching losses. Estimates of fundamental losses per cycle when the external field has a frequency of 10 gigahertz were made. Transport losses are primarily acoustic, estimated at below 10-27 joules per cycle.
Dielectric losses vary widely depending on the exact crystal symmetry, with even the equation describing the loss being different for different symmetry groups. A rough estimate for the fundamental dielectric loss for diamond is below 10-35 joules per cycle per switch with an electric field strength of 108 volts/meter. While pure diamond will almost certainly have a different (and lower) loss than the highly structured complex logic circuitry that would be needed in an actual system, this calculation still suggests that energy dissipation from this source can be made very small (and can likely be reduced to below 10-27 joules per cycle, the dissipation estimated from other loss mechanisms). Further research on the dielectric loss is required to better understand and minimize the losses that can be expected from this mechanism.
Switching losses below 10-27 joules per logic operation at switching speeds faster than 10-10 s should also be feasible. These losses involve both the excitation of electrons and their dissipative return to the ground state, as well as radiative losses resulting from the accelerations to which the switched electron is subjected during a switching operation.
These estimates support the conclusion that thermodynamically reversible single electron logic operations should eventually be able to achieve very low energy dissipations, very likely below 10-27 joules per logic operation at a temperature of 1 K and a speed of 10 gigahertz. The largest energy loss mechanism identified was acoustic radiation, which could be reduced by the use of dipoles. Even though the estimates of dielectric loss and radiative loss were small there is substantial uncertainty in their values: further analysis seems appropriate.
In the present proposal simply increasing the clock frequency would result in faster switching times, though at the cost of increased energy dissipation and error rate since no attempt was made to keep track of phase information. The switching time was simply made long compared with the time of a single oscillation of the first excited state of the switched electron, thus allowing use of the adiabatic approximation. If the switching time were ~h/H and if we computed the wave function of the switched electron as it left the switch, it should be possible to "catch" the switched electron in an appropriately designed potential. Switching speeds of ~10-14 s with minimal energy dissipation and error rates should be feasible using this approach. With lateral movement of the switched electron of ~10-8 m we have electron speeds of ~106 m/s (~.003 c), accelerations of ~1020 m/s2, forces of ~10-11 N, and an electron-electron separation (in vacuum) of ~5 × 10-9 m. Relativistic effects and materials limitations should not pose fundamental limitations for operations at these speeds.
A switching speed faster than ~10-14 s in the context of the current proposal would require creating a larger switching potential by, for example, using more than one switching electron (i.e., a charge packet). Use of a suitable charge packet would also permit reliable room temperature operation. More detailed analysis of rapid switching would be of interest.