CA2003134C - Solid state, quantum mechanical, electron and hole wave devices - Google Patents

Abstract

Solid state, quantum mechanical, electron or hole wave devices (100) formed from superlattice structures (120) provide, among other things, energy selectivity for substantially ballistic electron or hole wave propagation in the superlattice structures at energies above the superlattice potential energy barriers. The devices are designed by transforming designs for optical, thin film filter into designs for the semiconductor devices. The transformation, for electron wave devices, includes mapping the optical phase index into a solid state phase index which is proportional to the square root of the product of the electron kinetic energy and the electron effective mass and mapping the optical amplitude index into a second solid state amplitude index which is proportional to the square root of the electron kinetic energy kinetic energy divided by the electron effective mass. One embodiment is a tunable, biased, semiconductor superlattice, electron interference filter/emitter (1100) which can serve, for example, as a hot emitter in a ballistic transistor. Another embodiment is a semiconductor, quantum well, electron or hole slab waveguide (2100) which includes a substrate semiconductor layer (2200), a film semiconductor layer (2201), and a cover semiconductor layer (2202).

Description

` 200:~34 SOLID STATE, QUANTUM MECHANICAL, ELECTRON AND HOLE WAVE DEVICES
Backqround of the Invention The present invention pertains to solid state, quantum mechanical, electron and hole wave devices and methods for fabricating them and, in particular: (a) to solid state quantum mechanical electron and hole wave devices such as, without limitation, low pass filters, high pass filters, narrow band and wide band notch filters, narrow band and wide band bandpass filters and impedance transformers; (b) to voltage-biased, semiconductor, superlattice structures such as, without limitation, tunable, voltage-biased, electron wave, interference filter/emitters that can serve, for example, as a hot emitter in a ballistic transistor; and (c) to semiconductor, quantum well, electron and hole waveguides.
Recent progress in semiconductor growth technologies, particularly in molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD), enahle those of ordinary skill in the art to grow multilayered superlattice structures with precise monolayer compositional control. For example, successively grown layers of narrow and wide band gap semiconductor materials such as GaAs and Gal_xAlxAs have been produced and widely used to provide multiple quantum well structures. In fact, there are references in the prior art which are concerned with the use of these superlattice structures in resonant tunneling, superlattice/multiple quantum well devices. Specifically, in such devices, a superlattice is formed by growing successive layers of narrow and wide band gap semiconductor material epitaxially and the materials and the widths of the layers in these devices are chosen so that quantum states which arise from spatial quantization effects in adjacent wells become coupled. Further, in such devices, the interaction of these coupled states leads to the formation of minibands of allowed energies through which carriers can tunnel.

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._ Most of the above-described resonant tunneling superlattice devices disclosed in the prior art comprise a single quantum well, two barrier structure --see, for example, an article by T. Inata, S. Muto, Y. Nakata, S. Sasa, T. Fujii, and S. Hiyamizu, Jap. J. Appl. PhYs~ 26, L1332 (1987); an article by M. A. Reed, J. W. Lee, and H-L. Tsai, Appl. Phys. Lett., 49, 158 (1986); and an article by S. Y.
Chou and J. S. Harris, Appl. Phys. Lett., 52, 1442 (1988)--and such devices are of great interest as high frequency microwave oscillators --see, for example, an article by T. C.
L. G. Sollner, W. D. Goodhue, P. E. Tannewald, C. D. Parker, and D. D. Peck, Appl. Phys. Lett., 43, 588 (1983) and an article by T. C. L. G. Sollner, E. R. Brown, W. D. Goodhue, and H. Q. Lee, APPl. Phys. Lett., 50, 333 (1987). Recently, however, resonant tunneling through a multiple layer structure consisting of three wells and four barriers has been demonstrated in a GaAs/AlGaAs material system --see, for example, an article by C. J. Summers, K. F. Brennan, A.
Torabi, and H. M. Harris, Appl. Phys. Lett., 52, 132 (1988).
Further, these structures have potential use as high energy injectors for electroluminescent devices, photodetectors, and fast ballistic transistors --see for example, an article by C. J. Summers and K. F. Brennan, APPl. Phys. Lett., 48, 806 (1986); an article by K. F. Brennan and C. J. Summers, J.
Appl. PhYs.l 61, 5410 (1987); and an article by K. F. Brennan and C. J. Summers, IEEE J. Ouantum Electron,, QE-23, 320 (1987).
In addition to the above, the following prior art references: an article by T. Nakagawa, H. Imamoto, T.
Sakamoto, T. Kojima, K. Ohta, and N. J. Kawai, Electronics Letters, Vol. 21, No. 19, 882 (1985) and an article by T.
Nakagawa, N. J. Kawai, and K. Ohta entitled "Design Principles for CHIRP Superlattice Devices," Superlattices and Microstructures, Academic Press Inc. Limited, Vol. 1, No. 2, 1985, pp. 187-192; disclose the use of superlattices to 200:~13~

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provide miniband and forbidden energy bands at carrier energies above the barrier heights in order to produce negative differential resistance effects or to act as low-transmissivity blocking contacts.
In further addition to the above, there is presently great interest in the art in providing devices which exhibit high speed operation. Specifically, a major factor affecting the speed of semiconductor devices is the transit times of electrons from the input to the output terminals. It is expected that if one can provide electrons which pass through the semiconductor without any scattering events, namely by "ballistic" or "collisionless" motion, then the transit time will be minimized and the potential speed of the devices will be maximized. The possibility of ballistic motion in semiconductor materials has recently been provided by experimental results in GaAs, which results have been disclosed in an article which is not prior art by M. Heilblum entitled "Ballistic Electrons and Holes Observed in a Semiconductor," oPticS News, 1988, pp. 13-16. As such, it is expected that when the length of the region to be traversed is on the same order as the electron mean free path (mfp), a sizable fraction of the electrons will traverse it ballistically. For example, although the mfp in silicon is on the order of 100 Angstroms (A), the mfp for electrons in GaAs is approximately 10 times greater.
In the interest of investigating the efficacy of fabricating such ballistic electron devices, experiments have been described in the prior art in which a GaAs layer is sandwiched between two layers of an alloy of AlGaAs. AlGaAs is reported to be a suitable material for use therein because it has the same lattice constant as GaAs and, as a result, it can be grown epitaxially thereon. In addition, further reported experiments have shown that ballistic hole motion also occurs in GaAs, albeit at a lower fraction than that which occurs for electron motion due to the peculiar band structure of the valence band of GaAs.
In light of the above, there is a need in the art for electron and/or hole filter devices, such as low pass, high pass, notch and bandpass filters which can be used to fabricate solid state devices requiring energy selectivity such as, for example, electroluminescent devices, photodetectors, ballistic transistors and filter/emitter structures which can serve as high energy electron injectors in these devices and electron or hole waveguide devices for use in fabricating analogs of integrated optical devices.
Summary of the Invention The invention in one aspect provides for solid state quantum mechanical electron and hole wave devices each comprising a superlattice structure characterized in that the superlattice structure comprises a multiplicity of adjacent layers of semi-conductor material, each of which layers has a potential energy barrier and supports substantially ballistic electron and hole transport at energies above the potential energy barrier of the layer.
Another aspect of the invention provides for solid state, quantum mechanical, electron and hole wave filter/emitters which each comprise a superlattice structure which is comprised of a multiplicity of adjacent layers of semi-conductor material, each of which layers has a potential energy barrier and supports substantially ballistic electron or hole transport at energies above the potential energy barrier of the layer and means for applying a bias potential energy to the superlattice structure wherein the potential energy barriers, electron and hole effective masses and thicknesses of the layers of the superlattice structure are predetermined as if a predetermined bias potential has been applied to the applying means so that the application of the predetermined bias potential to the applying means causes the filter/emitter to function as a filter/emitter for electrons and holes having kinetic energies in a predetermined range.
Still further the invention provides a method for fabricating an electron wave, semiconductor device comprising a superlattice structure which supports substantially ballistic electron transport at energies above the superlattice potential energy barriers, which method comprises the steps of forming an epitaxial superlattice structure comprised of a predetermined number of layers of semiconductor materials, characterized in that the thickness and the composition of each of the layers are chosen by utilizing an electromagnetic optical wave device design whose parameters are determined in accordance with optical design methods and converting the parameters of the optical device design into the thickness and composition of each of the layers of the superlattice structure by mapping the optical phase index of refraction into a first solid state index of refraction for phase quantities which is proportional to the square root of the product of the electron kinetic energy and the electron effective mass and by mapping the optical amplitude index of refraction into a second solid state index of refraction for amplitude quantities which is proportional to the square root of the electron kinetic energy divided by the electron effective mass.
Further still the invention provides a method for fabricating a biased, electron wave, semiconductor, filter/emitter comprising a superlattice structure which supports substantially ballistic electron transport at energies above the superlattice potential energy barriers, which method comprises the steps of forming an epitaxial superlattice structure comprised of a predetermined number of layers of semiconductor materials, characterized in that the thickness and the composition of each of the layers are determined in accordance with an iterative method which utilizes as an initial estimate an electromagnetic optical wave device design whose parameters are determined in accordance with optical design methods and converting the parameters of the optical device design into the thickness and composition of each of the layers of the superlattice structure by mapping the optical phase index of refraction into a first solid state index of refraction for phase quantities which is proportional to the square root of the product of the electron kinetic energy and the electron effective mass and by mapping the optical amplitude index of refraction into a second solid state index of refraction for amplitude quantities which is proportional to the square root of the electron kinetic energy divided by the electron effective mass.
More particularly various embodiments of the present invention solve the above-identified need in the prior art and relate to three categories of embodiments: (1) unbiased devicesi (2) biased devices and (3) waveguides.
Unbiased Devices:
Embodiments of the present invention solve the above-identified need in the art by providing solid state quantummechanical electron and hole wave devices which can provide, among other things, energy selectivity. Specifically, embodiments of the present invention comprise solid state: low pass filters; high pass filters; narrow band and wide band notch filters; narrow band and wide band bandpass filters; impedance transformers which are analogous to optical antireflection coatings and high reflectance devices which are analogous to mirrors.
Embodiments of the present invention comprise multilayer superlattice structures comprised o~ materials which support substantially ballistic electron transport. When electrons are injected into such structures at energies above the corresponding potential energy barriers, the structures provide electron interference effects which are exactly analogous to those which occur in electromagnetic wave propagation 20 in dielectric films. These embodiments are 200313~

designed in accordance with the inventive method by utilizing a first solid state index of refraction for phase quantities which is proportional to the square root of the product of electron kinetic energy and electron effective mass and by utilizing a second solid state index of refraction for amplitude quantities which is proportional to the square root of electron kinetic energy kinetic energy divided by electron effective mass. The indices provide an exact analogy between a quantum mechanical electron wave and an electromagnetic 0 optical wave wherein the electron wavevector:
k = [2m (E - V)]1/2/~ (1) is analogous to the optical wavevector and the electron wave amplitude refractive index:
ne(amplitude) oc[(E - V)/m ]1/2 (2) is analogous to the optical index of refraction. These indices are used in expressions for reflectivity and transmissivity at a boundary which are well known to those of ordinary skill in the art from electromagnetic design to provide a method for transforming existing optical interference filter designs into designs for inventive semiconductor devices. For example, these mappings between optical quantities and solid state quantities may be used to transform well known optical designs into designs for analogous solid state filter devices which have Butterworth, Chebyshev, elliptic function, or other well known filter characteristics. In particular, one embodiment of an inventive solid state electron wave device comprises a solid state analog of a Fabry-Perot optical interference filter which is fabricated from alloys of AlGaAs and GaAs. Further, the inventive solid state filter devices can be incorporated monolithically into transistor structures in order to increase their speed.
The efficacy of the mapping between electromagnetic optical waves and quantum mechanical electron waves depends on the existence of ballistic electron transport in the solid Z00:~134 state materials, i.e., wherein electrons travel through the solid state materials without being scattered by deviations from crystalline perfection. The ballistic electrons have energies above the potential barriers in the solid state materials and exhibit quantum mechanical plane wave behavior.
Further, since these plane waves maintain their phase through the device, these coherent waves will refract, reflect, interfere, and diffract in a manner which is analogous to the behavior of electromagnetic waves traveling through dielectrics.
Doping of semiconductors is not important for the embodiments of the present invention, however, it preferred to exclude doping within the active region of the device in order to avoid scattering within the materials. This provides a further advantage for the inventive devices because the absence of doping makes them easier to fabricate.
Although we have discovered that electron wave propagation at energies above the potential barriers can be mathematically described by a mapping between quantum mechanical electron waves in semiconductors and electromagnetic optical waves in dielectrics, semiconductor superlattice interference filter designs, for example, cannot merely be copies of thin-film optical filter designs. This is because, although optical designs which may be realized in nature are constrained by the available material indices of refraction, the design of the analogous semiconductor devices, such as, for example, superlattice interference filters, will be constrained by the fact that: (a) the thicknesses of layers in the superlattice structures are restricted to be integer multiples of monolayer thicknesses and (b) the requirement of substantially collisionless transport for the carriers limits the usable composition ranges of the materials. This latter constraint occurs because the requirement of collisionless transport often precludes using material compositions which have indirect 20031~4 _ band gaps. Nevertheless, as should be clear to those of ordinary skill in the art, one may use a trial and error method of determining which designs are physically realizable. However, a preferred embodiment of the inventive method which is described in the Detailed Description comprises a systematic method for determining superlattice designs which meet the appropriate physical constraints.
Biased Devices:
Embodiments of the present invention solve the above-identified need in the art by providing a biased, semiconductor, superlattice, tunable, electron interference filter/emitter which can serve, for example, as a hot electron emitter in a ballistic transistor. In particular, an embodiment of the present invention comprises a biased, semiconductor, superlattice filter/emitter which provides energy selectivity for substantially ballistic electron wave propagation at electron energies above the superlattice potential barriers. Further, the layers of the biased, superlattice structure comprise alternately high and low electron refractive indices wherein each layer is a multiple of a quarter or half of an electron wavelength in thickness and wherein quantum well barrier widths are adjusted in the direction of emission to provide the desired energy selectivity.
Embodiments of the inventive biased, semiconductor, superlattice filter/emitter are designed, in accordance with the inventive method, by transforming optical, thin-film interference filter designs which are determined in accordance with existing optical interference filter design methods into inventive semiconductor devices as discussed above with respect to unbiased devices. However, in this case of biased devices, V in eqns. (1) and (2) above, is a function of position in the semiconductor materials because of the applied bias.

200~ 4 _ Wavequides:
Embodiments of the present invention solve the above-identified need in the art by providing semiconductor, quantum well, electron or hole slab waveguides. For example, such electron waveguides should be useful in high-speed circuitry and as a central component in electron or hole guided wave integrated circuits. Specifically, an electron slab waveguide is comprised of a substrate semiconductor layer, a film semiconductor layer, and a cover semiconductor layer, wherein the semiconductor layers provide substantially ballistic transport for electrons and wherein the thicknesses and compositions of the semiconductor layers are determined in accordance with the inventive method to provide a potential well.
In particular, in accordance with the present invention, electron waveguide modes exist for electron energies in the well and for electron energies above one or both OL the po~ential energy barriers of the substrate layer and the cover layer, respectively. Further, in contrast to the behavior of electromagnetic guided waves which only have a lower-energy cutoff due to dispersion, each electron waveguide mode also has an upper-energy cutoff wherein an electron wave is refracted into the substrate layer and/or the cover layer.
Brief Description of the Drawinq A complete understanding of the present invention may be gained by considering the following detailed description in conjunction with the accompanying drawing, in which:
FIG. 1 shows, in pictorial form, an electron, superlattice, interference filter fabricated in accordance with the present invention;
FIG. 2 shows, in pictorial form, the energy level diagram and the material composition of the electron, superlattice, interference filter of FIG. 1;

.

FIG.3 shows, in pictorial form, the index refraction diagram of an optical, thin film, interference filter which is the optical analog of the electron, superlattice, interference filter of FIG. l;
5FIG. 4 shows, in pictorial form, the transmissivity of the electron, superlattice, interference filter of FIG. l;
FIG. 5 shows, in pictorial form, the energy level diagram and the material composition of a biased, electron wave, superlattice, interference filter/emitter fabricated in 10accordance with the present invention;
FIG. 6 shows, in pictorial form, the transmissivity of the electron, superlattice, interference filter of FIG. 5;
FIG. 7 shows, in pictorial form, the energy level diagram and the material composition of an asymmetric, 15quantum well, slab waveguide fabricated in accordance with the present invention;
FIG. 8 is a plot of the electron guided mode propagation constant as a function of total electron energy which shows the regions of evanescent modes, guided modes, 20substrate modes, and radiation modes for a quantum well slab waveguide comprised of a GaO 85Alo 15As substrate layer, a GaAs film layer, and a GaO 70Alo 30As cover layer, along with mode dispersion curves for the fundamental mode, Mo~ for various film layer thicknesses; and 25FIG. 9 is a plot of the wavefunction Uv for the Mo mode of a slab waveguide comprised of the GaAlAs material system having a 10 monolayer thick GaAs film layer for various electron energies and appears with FIG. 7.
To facilitate understanding, identical reference 30numerals are used to designate elements common to the figures.
Detailed Description The following describes embodiments of the present invention which relate to three categories of embodiments:
35(1) unbiased devices; (2) biased devices; and (3) waveguides.

, ~

20031.~34 .

UNBIASED DEVICES
FIG. 1 shows, in pictorial form, an electron wave, superlattice, interference filter 100 fabricated in accordance with the present invention. Inventive filter 100 provides a narrow, bandpass, transmission filter for specific incident electron kinetic energies which are greater than the heights of the potential barriers of the materials which comprise layers 101-109 of superlattice 120 and input layer 131 and output layer 132. As will be described in detail below, the thicknesses of layers 101-109 and the compositions and heights of the potential barriers of the materials comprising layers 101-109 are determined in accordance with the inventive method by utilizing a mapping between quantum mechanical electron waves in semiconductors and electromagnetic optical waves in dielectrics. The inventive method advantageously uses this mapping to enable one to apply existing thin film optical design techniques and existing optical designs to design corresponding solid state quantum mechanical electron wave devices.
FIG. 2 shows, in pictorial form, the energy level diagram and the material composition of electron wave, superlattice, interference filter 100 of FIG. 1. As shown in FIG. 2, input and output layers 131 and 132, respectively, are comprised of GaO 55Alo 45As. Superlattice 120 is comprised of quarter-wavelength GaAs layer 101, quarter-wavelength GaO 55Alo 4sAs layer 102, quarter-wavelength GaAs layer 103, quarter-wavelength GaO ssAlo 45As layer 104~ half-wavelength GaAs layer 105, quarter-wavelength GaO 55Alo 45As layer 106, quarter-wavelength layer GaAs 107, quarter-wavelength GaO 55Alo 45As layer 10$, and quarter-wavelength GaAs layer 109, where the term quarter-wavelength and half-wavelength will be defined below. Further, for filter 100 of FIG. 1, layers 101, 103, 107, and 109 are each comprised of six (6) monolayers of GaAs and are, therefore, 16.9599 A
thick, whereas layers 102, 104, 106 and 108 are each 2no.~ 4 comprised of nine (9) monolayers of GaO 55Alo 45As and are, therefore, 25.4398 A thick. In this embodiment, we have taken a monolayer of both GaAs and GaO 55Alo 45As to have a thickness of 2.82665 A. Still further, for filter 100 of FIG. 1, the thickness of layer 105 is twice that of layer 101 .
FIG. 4 shows, in pictorial form, the transmissivity of electron wave, superlattice, interference filter lOo of FIG. 1. The pass electron kinetic energy, determined in the manner explained below by reference to the kinetic energy of an electron input into GaO 55Alo 45As layer 131, is 0.139 eV
and the passband has a full width at half maximum (FWHM) of 0.003 eV, which passband is only 2.2% of the pass electron kinetic energy. Further, it is important to note that the inventive filter advantageously provides a transmission at maximum which is substantially equal to 100%.
We will now describe the inventive method which utilizes a quantitative mapping between quantum mechanical electron waves in semiconductor materials and electromagnetic optical waves in dielectrics. The mapping is utilized, as explained below, to translate existing optical design methods and existing optical designs into methods and designs for analogous solid state devices. In accordance with the inventive method, the mapping comprises: (1) mapping optical phase effects which depend on path differences by using an "electron wave phase refractive index" ne(phase) that is proportional to the square root of the product of the electron effective mass and the electron kinetic energy, i.e., tm (E-V)]1/2 and (2) mapping optical amplitude effects such as transmissivity and reflectivity by using an "electron wave amplitude refractive index~ ne(amplitude) that is proportional to the square root of the ratio of the kinetic energy to the effective mass, i.e., [(E-V)/m ]1/2 The mapping utilizes the discovery that quantum mechanical electron waves in semiconductors and 2no.~

electromagnetic optical waves in dielectrics exhibit transmission, reflection, interference, and diffraction characteristics that are exactly analogous to each other. As a result, existing optical device designs now have electron and hole wave device counterparts. This mapping is derived by recognizing that: (a) continuity of the quantum wave function across a potential energy boundary is analogous to the continuity of the tangential component of the electric field across a boundary between dielectrics and (b) conservation of electron probability current normal to a potential energy boundary is analogous to conservation of power flow normal to a boundary between dielectrics. The mapping of the inventive method may be restated as follows for using existing optical design methods and existing optical designs: (1) use the following electron wavevector k = [2m*(E - V)]1/2/~ in place of the optical wavevector and (2) use the following electron wave amplitude refractive index [(E - V)/m ]1/2 in place of the optical index of refraction in expressions for reflectivity and transmissivity at a boundary.
It is important to note that the above-described mapping applies to the case of electron waves in semiconductor materials, which case depends, in turn, on the fact that: (1) the electrons have energies above the potential barriers in the semiconductor materials --note electron energy E in FIG. 2-- and (2) the electrons exhibit ballistic or collisionless motion in the semiconductor materials. In addition, note the following important factors concerning the mapping of the inventive method: (1) even though electromagnetic optical waves involve polarization and the mapping for amplitude effects is a single parameter which has dimensionality, there appears to be no inconsistency because only dimensionless ratios of the electron wave amplitude refractive indices appear in transmissivity and reflectivity expressions for electron waves; (2) both the 2no~ 4 _ 13 phase effect and amplitude effect electron wave refractive indices exhibit "normal" dispersion, i.e., they increase with decreasing wavelength or increasing energy; and (3) although semiconductor materials (a) may have nonparabolic bandstructure in terms of E vs. k, i.e., energy vs. momentum, and (b) may have band structures which vary with a particular direction of electron wave propagation in the material, these effects may be incorporated into the inventive method by using an energy dependent, anisotropic effective mass m .
Thus, even though the allowed wavevector surface is no longer spherical in the presence of anisotropy, all of the inventive design methods set forth herein still apply, provided the energy dependent anisotropic effective mass is used in the analysis.
In light of the above-described discovery, it has been determined that inventive electron wave, superlattice devices such as interference filters share common characteristics with thin-film optical interference filters.
As a result, it is useful to review some of the primary properties of these optical filters and, in so doing, we will, at the same time, be reviewing characteristics of inventive electron wave, superlattice, interference filters.
A simple type of narrow bandpass, optical, interference filter is a Fabry-Perot filter. It is comprised 2S of a half-wavelength layer, frequently referred to as a "spacer" in the optical literature, which is sandwiched between reflectors. In the case of an all-dielectric Fabry-Perot filter, the reflectors are stacks of high index, designated H, and low index, designated L, quarter-wavelength layers. The FWHM of the bandpass of this type of filter can be reduced by increasing the reflectivity at the boundaries between the-layers. This may be accomplished by increasing the ratio of the high index of refraction, nH, to the low index of refraction, nL. Furthermore, for a given number of layers, the higher reflectances occur with the high index, H, 2~0.~
_ 14 layers on the outside boundaries of the filter. The half-wavelength, resonant layer at the center of the filter may be of high index of refraction, nH, or low index of refraction, nL, material. Thus, there are two basic types of all-dielectric Fabry-Perot interference filters. In the optical literature, these are symbolically represented as [HL]NHH[LH]N and H[LH]NLL[HL]NH where H and L represent quarter-wavelength layers of high and low index of refraction materials, respectively, and N represents the number of repetitions of the layer-pair type indicated in square brackets.
Other important characteristics of all-dielectric interference filters which are well known to those of ordinary skill in the art, which characteristics, in accordance with the inventive method also apply to the analogous, inventive solid state electron wave filters, are:
(CHl) the maximum transmittance of the filter is 100%; (CH2) the maximum transmittance occurs at the wavelength for which (a) the spacer layer is a half-wavelength thick, as measured in that material, and (b) the reflected layers are a quarter-wavelength thick, as measured in those materials, which wavelength will be referred to below as the pass wavelength;
(CH3) the FWHM and the finesse are controlled by the number of surrounding quarter-wavelength layers, i.e., the FWHM is decreased and the finesse is increased as further quarter-wavelength sections are added; (CH4) the transmittance characteristic is symmetric about the pass wavelength when the transmittance characteristic is plotted as a function of the reciprocal of the wavelength, as measured in the material surrounding the filter; (CH5) a proportional change in the thicknesses of all layers produces a simple displacement of the transmittance characteristic plotted as a function of the reciprocal wavelength; (CH6) if the thicknesses of all layers are increased by an odd integer factor, a passband will occur at the original pass wavelength and it will have a decreased 200.~ 4 ~ , FWHM; (CH7) as the angle of incidence upon the filter is increased, the pass wavelength is tuned to shorter wavelengths; (CH8) the transmittance characteristic is relatively insensitive to variations in the reflectivities and thicknesses of the layers; (CH9) normal dispersion causes a narrowing of the FWHM; and (CH10) the filter is effective over only a limited range since sidebands necessarily occur on either side of the passband.
Specific nomenclature well known to those of ordinary skill in the art when referring to such filters is given as follows: (1) the range from the nearest passband peak below the pass wavelength to the nearest passband peak above the pass wavelength is called the free spectral range, FSR; (2) the finesse is equal to FSR/FWHM; and (3) the resolving power is equal to pass wavelength/one-half FWHM.
Using the above-described mapping we can determine the characteristics of a many-boundary semiconductor superlattice system. This is done, as shown in Appendix I, by applying the chain-matrix approach commonly used in electromagnetics and by substituting the electron wavevector given by the inventive mapping for phase quantities for the optical wave vector and by substituting the electron wave amplitude refractive index given by the inventive mapping for indices of refraction in the expressions for the reflectivity and transmissivity at a boundary.
The following example shows how to translate an optical design into a solid state design. In this example, an optical thin film design that appears in a book by A.
Thelen entitled ~'Physics of Thin Films," vol. 5, edited by G.
Hass and R. E. Thun, Academic Press New York, eds., 1969, in the chapter which starts at p. 47 is translated into a design for an electron wave filter. The optical design is an eleven-layer structure which exhibits a FWHM bandpass of 2.2%
of the design pass wavelength. In the notation of optical thin film design, the optical filter is designated 2no.~

-1.0 HL HH LHLHL HH L'H 1.0; where 1.0 indicates air for the input and output regions, H indicates a quarter-wavelength thickness layer of high index (as measured in the medium), and L indicates a quarter-wavelength thickness layer of low index (as measured in the medium). Thus, the notation HH
signifies a half-wavelength thickness layer of high index.
For the optical design, nH = 4 0, nL = 1.35, and nL, = 1.83.
For a pass wavelength of 1.00 ~m, the physical thicknesses of the layers for the optical filter are given in Table I.
The corresponding electron wave, superlattice filter design is obtained in accordance with the inventive mapping set forth above in the following manner. First, calculate the electron pass kinetic energy (E - V0) in the input region, m=0, associated with the desired electron wave pass wavelength lame 0 in the input region from the following equation:
(E - V0) = h2/2mQ lame o2 (1) where mO* is the electron effective mass for the input or m=0 region. Second, calculate a scaling factor D to convert the quantity [(E - Vm)/mm ]1/2 for layer m into ne m(amplitude) for the same layer from the following equation:
D = nO/[(E - V0)/mO ]1/2 (2) where nO is the refractive index of the input, m=0, region of the optical design. Third, calculate for each m-th layer, the required value of (E - Vm) from the following equation:
(E - Vm) = (nm/D)2mm (3) where nm is the refractive index in the optical design for the m-th region and mm* is the electron effective mass~ for the m-th region. Fourth, calculate the thickness of the quarter-wavelength layers using the electron wave phase refractive index which gives the physical thickness of the m-th layer as:
dm = h/~25/2[mm (E - Vm)]l/2} (4) Then, by following this procedure, the optical thin film interference filter design is converted to an electron 2olo~ 4 -wave interference filter design. Selecting a pass wavelength of 100 A for the solid state filter, the resulting ~inetic energies (E - V) and thickness values are given in Table II, where for simplicity the electron effective masses are taken to be the free electron mass in all layers in this example.
If the calculated thicknesses are too small for reasonable practical fabrication of a semiconductor superlattice or the thicknesses are less than monolayer thicknesses, the thicknesses of all layers can be increased by an odd integer factor. Just as was described above for the case of an optical interference filter, this will cause the FWHM of the solid state filter to be decreased by the same factor and, as a result, the solid state filter will become a narrower band filter. For example, for the above described interference filter design, increasing the solid state layer thicknesses by a factor of 3 or 5 reduces the FWHM from 2.2% of the passband wavelength to 0.73% or 0.44%
of the pass wavelength, respectively. ~owever, the finesse of the filter, i.e., the ratio of the rejected bandwidth to the pass bandwidth, is also decreased when these larger thicknesses are selected. Further, it is also important to note that just like optical thin film interference filters, the inventive electron superlattice interference filter can also be continuously tuned to lower pass wavelengths and, hence, higher energies, merely by mechanically rotating the filter so as to increase the angle of incidence away from normal incidence.
The above described method was applied to design the inventive filter 100 shown in FIG. 1. Specifically, FIG.

3, shows the analogous optical interference wave filter design that corresponds to solid state filter 100 in FIG. 1.
As previously discussed, filter 100 comprises superlattice 100 formed from layers of GaAs and GaO 55Alo 45As, which superlattice 100 is surrounded by input and output GaO 55Alo 45As layers 131 and 132. The effective masses used 2~0~ 4 in the design of filter lO0 were taken from prior art literature to be m (GaAs)=0.067mO and m (GaO 55Alo 45As)=0.10435mO, where mO is the free electron mass. The conduction band edge energies used were V(GaAs)=0.0000 eV and V(GaO 55Alo 45As)=0.3479 eV and a monolayer thickness of both GaAs and GaO 55Alo 45As was taken to be 2.82665 A. As a result, for these materials, 6 monolayers of GaAs (d1 = 16.9599 A) and 9 monolayers of GaO 55Alo 45As (d2 = 25.4398 A) both closely correspond to quarter-wavelength layers at an electron wavelength of 101.652 A or an electron kinetic energy of 0.139456 eV as measured in the surrounding GaO 55Alo 45As.
The above described inventive structures can be fabricated using techniques well known to those of ordinary skill in the art such as, for example, molecular beam epitaxy. Further, in accordance with the present invention, one can change the pass kinetic energy with a judicious selection of the composition of the materials, such as, for example, the ratio of Ga to Al in the above described example. Thus, to obtain a specified pass kinetic energy, the design procedure is as follows: (1) for a material system of the F1_XGxH type, find values of x such that an integer number of monolayers of the material is equal to a quarter wavelength at the pass kinetic energy (E - V) as measured in the material; (2) select values of x which are separated as much as possible in order to maximize the reflectivity at each boundary (For example, in the above-described example, this is accomplished by choosing x = 0 and 0.45); and (3) a basic bandpass filter may then be constructed, as in optics, by sandwiching a half-wavelength layer between quarter-wavelength layers of alternating material types (The FWHM and the finesse are controlled by the number of surrounding quarter-wavelength layers, i.e., the FWHM is decreased and the finesse is increased as further quarter-wavelength sections are added). It should be clear to those of ordinary 2~0.~ 4 skill in the art that the inventive superlattice devices may be comprised of layers which have a multiplicity of differing material compositions in addition to the type of device shown, for example, in FIG. 1, comprises a superlattice formed from layers of only two different materials. The designs having layers of more than two different materials may result from the need to fabricate a solid state analog of a particular type of optical thin-film device.
In designing solid state devices in accordance with the present invention, one can choose an appropriate material composition which will provide the quarter-wavelength and half-wavelength layers required by a particular filter design by trial and error. However, we will now describe a preferred embodiment of the inventive method which provides a systematic method for choosing such compositions in a particular case. Consider a material system which forms a continuous set of alloys of the type F1_XGxH. The range of usable compositions is given by the range in x from 0 to xmax, for example. This occurs because there may be a possible transition at xmax from a direct energy bandgap material to an indirect energy bandgap material such as occurs in the case of Ga1_xAlxAs. Although there is no prohibition in principal against the use of indirect bandgap materials, they cannot be used where the transition between a direct and an indirect bandgap material or between two indirect bandgap materials requires a change in momentum.
This is because we are dealing with wave effects that occur in substantially collisionless motion.
In considering the systematic method for choosing material compositions, assume the solid state electron wave filter is comprised of three types of materials: (1) the materials which surround the superlattice, i=0 regions, have compositions xO; (2) the material in the high refractive index regions of the two-material superlattice, i=1 regions, have compositions x1; and (3) the material in the low 2nO.'?1.~

refractive index regions of the two-material superlattice, i=2 regions, have compositions x2. The monolayer thicknesses are rl and r2 for the i=l and i=2 regions, respectively. The electron potential energy in the three regions, as is well known in the art, is given by:
Vi = delEC = Axi i=o, 1, 2 (5) where delEc is the change in the energy of the conduction band edge and A is a constant. Further, it is also well known in the art that the electron effective mass in the three types of regions is given by:
m* = (B + Cxi)mO i=0,1,2 (6) where B and C are constants and mO is the free electron mass.
The electron kinetic energy in the i-th region is (E - Vi) = h2/2mi lami2. The total electron energy to be passed by the filter is designated Ep. The pass kinetic energy as measured in the various regions is thus:
Ep - Vi = h2/2mi (lamp)i2 i=o, 1, 2 (7) where (lamp)i is the pass wavelength as measured in the i-th region. The overall pass kinetic energy of the filter as measured in the material surrounding the superlattice is Ep - V0. This is the pass kinetic energy that is specified by the user and is thus the starting point in the design procedure. Using the above equations and solving for pass wavelength gives:
(lamp)i = h/~2mO[-ACxi2 + (CEp-AB)xi + BEp]}l/2i=0,1,2 (8) The thicknesses of the superlattice layers are designated di for i=1,2. These thicknesses must be integer multiples of the monolayer thicknesses ri. Further, these thicknesses must also be odd multiples of a quarter wavelength as measured in these regions. These constraints may be expressed as:
di = Piri - (2qi ~ l)(lamp)i/4 i=1,2 (9) where Pi is the integer number of monolayers for the i-th region and qi is a positive integer 1,2,3,... Using the 2nO.?1.~4 above two equations gives the following quadratic equation for composition xi:
ACxi2 + (AB - CEp)xi + (h2/32mO)[(2qi - 1)2/pi2ri2] - BEp = o (10) This equation can be solved by the well known formula for quadratic equations. In order to design a superlattice interference filter, at least two solutions for Xi must be found in the range from 0 to xmax. The smallest value of xi within this range will become xl, the composition of the high index material. The value of Pi that produces x1 becomes Pl~ the number of monolayers of type 1 material used to make a quarter-wavelength layer. Similarly, the largest value of xi within this range will become x2, the composition of the low index material. The value of Pi that produces x2 becomes P2, the number of monolayers of type 2 material used to make a quarter-wavelength layer.
To allow the broadest range of solutions, V0 is set to Vmax. For a specified pass kinetic enrgy (E - V0), the value of Ep is then determined. Further, to minimize the total thickness of the filter, qi is initially set equal to unity. Then the quadratic equation is repetitively evaluated for Pi = 1,2,3, ... until all of the positive real roots in the range from 0 to xmax are found. If only one root or no roots are found, then the procedure must be started over with changed parameters. The quantities that can be varied are the integer qi, the surrounding material composition xO which changes Ep, and the crystallographic ~irection of growth which changes ri.
After x1, P1, x2, and P2 have been determined as described above, the other parameters of the filter can be calculated. The potential energies Vi, the effective masses mi , the electron wavevector magnitude ki, and the electron wave amplitude refractive index ne(amplitude)i can be computed in accordance with the above-described equations for each type of region.

2no.~ 4 ` 22 -As an example, consider the following example for the Gal_xAlxAs material system. This is an advantageous material system because all compositions are lattice matched for these alloys. For growth along the [100] direction, the S monolayer thickness is r=rl=r2=2.282665 A. The material is a direct gap semiconductor for x less than or equal to 0.45 and, as a result, this represents the usable composition range. Further~ for Gal_xAlxAs, A=0.77314 eV, B=0.067 and C=0.083.
As an example, to design a Gal_xAlxAs superlattice interference filter with a pass kinetic energy of 0.20 eV
such as might be useful for an emitter in a high speed ballistic transistor, the following calculations are performed. Let xO = xmax = 0.45. Thus V0 = 0.347913 and, lS since Ep - V0 = 0.20 eV, then Ep = 0.547913 eV. Letting qi =
1, the composition xi is evaluated for Pi = 1,2,3,... until all of the positive real roots are found in the range of o to 0.45. For the present case there are two roots, i.e., x2 =
0.3984 corresponding to P2 = 6 and xl = 0.2063 corresponding to Pi = 7- The smaller value of xi is designated xl and the larger value x2. The thickness of the GaO 79Alo 21As quarter-wavelength layer is dl = plr = 19.7866 A, seven (7) monolayers. The thickness of the GaO.60AlO.40As quar e wavelength layer is d2 = P2r = 16.9599 A, six (6) monolayers.
The electron effective masses in the three regions are calculated to be mO* = 0.10435mO, m1* = 0.084126mO, and m2 0.10007mO. The pass kinetic energies in the three regions are Ep - V0 = 0.2000 eV, Ep - Vl - 0.3884 eV, and Ep - V2 = 0.2399 eV. The electron wave amplitude refractive indices normalized to the surrounding i=o regions are ne(amplitude)0 = 1.000000, ne(amplitude)l = 1.552027, and ne(amplitude)2 = 1.118372. For a 13-layer Fabry-Perot interference filter of the form tHL]3HH[LH]3, these calculated material characteristics produce a filter having a pass band of 0.20 eV and a FWHM of 15.4 meV.

2nO.~31 .~4 Repeating the above-described procedure, Ga1_xAl~s superlattice filters were designed for pass kinetic energies for 0.14 eV up through 0.20 eV, the range of energies potentially most useful in ballistic transistors. The positive real roots are shown in Table III for 6 through 10 monolayer thicknesses. Roots must be in the range from 0 to 0.45. At the 0.14 eV low end of this energy range, there are essentially four roots. At the 0.20 eV high end of the energy range, there are two roots.
There is considerable flexibility in the design of semiconductor superlattice interference filters. For example, other odd multiples of a quarter wavelength may be used qi =2,3,4, ..., the surrounding material can be changed to alter V0, and other crystallographic growth directions may be used to alter ri.
Clearly, those skilled in the art recognize that further embodiments of the present invention may be made without departing from its teachings. For example, it is within the spirit of the present invention to provide a wide variety of hole wave devices as well as electron wave devices. In addition, it is within the spirit of the present invention that a wide variety of electron or hole wave devices which use electron or hole wave propagation above the potential barrier can be fabricated which are analogous to electromagnetic optical wave devices. Further, in accordance with the inventive method, such devices can be fabricated using the existing methods for determining optical designs and using the inventive mapping described above.
Specifically, such devices include low pass filters, high pass filters, notch filters (narrow band and wide band), bandpass filters (narrow band and wide band), impedance transformers (antireflection devices), and high reflectance surfaces (mirrors). In addition, the filters can have Butterworth (~xi~lly flat), Chebyshev, elliptic function, or other types of characteristics which are well known in the 2~0~

art. Further in addition, such inventive devices may be used to provide monoenergetic electron sources for an entire class of devices such as electroluminescent devices, photodetectors, and ballistic transistors. Still further in addition, such inventive devices may be used to aid in contolling, shaping and filtering freespace electron beams to provide electron spectrometers, electron lithography and electron diffraction analysis of crystals.
In addition to the above, in accordance with the present invention, one can guide electrons or holes by, for example, injecting them into a semiconductor layer which is sandwiched between superlattices which have been designed, in accordance with the techniques described above, to provide total reflection.
In terms of nomenclature, it should be clear to those of ordinary skill in the art that references to electron energies being above the potential barriers, correspond to energies, as shown in FIG. 2, which are above the conduction band. Further, it should also be clear to those of ordinary skill in the art that similar references for holes correspond to energies which are below the valence band.

200.~

BIASED DEVICES
FIG. 5 shows, in pictorial form, the energy level diagram and the material composition of voltage-biased, electron, superlattice, interference filter/emitter 100 fabricated in accordance with the present invention.
Inventive filter/emitter 100 comprises M layers, layers 2001 - 200M. Layers 2001 - 200M are surrounded by bulk semiconductor material layers 11001 and 11002 and a predetermined bias potential Vbias/q is applied to filter/emitter 100 by placing a voltage source (not shown) across layers 2001 - 200M/ q being the electronic charge.
As shown in FIG. 5, electrons 250 are injected into filter/emitter 100 from layer 11001. Further, only those electrons having kinetic energy (KE)in a narrow spectral band around pass energy Ep --which passband energy Ep is above the potential barriers of layers 2001 - 200M-- traverse filter/emitter 100 and are emitted into layer 11002. Still further, such electrons are emitted into layer 11002 with an output kinetic energy (KE)oUt which is larger than input kinetic energy (KE)in-In accordance with the present invention, because(KE)oUt > (KE)in, inventive filter/emitter 100 provides an emitter function for electrons having an incident kinetic energy of (KE)in. As one can readily appreciate, emission of electrons from the inventive device at a kinetic energy which higher than their input kinetic energy is achieved by the application of a bias potential Vbias/q to a filter whose filter properties are affected and determined by the resulting bias potential energy Vbias. In other words, filter/emitter 1100 provides a narrow bandpass transmission filter/emitter for electrons having a specific incident kinetic energy, which electrons have a total energy which is greater than the heights of the potential barriers of the materials which comprise layers 2001 - 200M of the superlattice.

2no~ 4 As will be described in detail below, the thicknesses of layers 2001 - 200M~ the compositions of the materials comprising layers 2001 - 200M~ and the heights of the potential barriers of layers 2001 - 200M are determined in accordance with the inventive method by utilizing the above-described mapping between quantum mechanical electron waves in semiconductors and electromagnetic optical waves in dielectrics. The inventive method advantageously uses this mapping to apply thin film optical designs which were designed using existing optical filter design techniques to design analogous solid state quantum mechanical electron wave filter/emitter devices.
The basic structure for inventive filter/emitter 100 is that of an unbiased superlattice electron wave interference filter like those disclosed above in the section covering "Unbiased Devices." Specifically, suitable unbiased superlattice electron wave interference filters are determined, by analogy to optical filters, to be comprised of successive layers of odd and even multiples of electron quarter wavelengths, for example, the narrow bandpass optical interference Fabry-Perot filter described above.
As further shown in FIG. 5, the jth barrier or quantum well has a thickness which is denoted by dj and, at zero bias, a potential energy which is denoted by Vj.
Although these are not limitations of the present invention, to make it easier to understand the manner in which this embodiment of the present invention operates: (a) surrounding layers 11001 and 11002 are chosen to have the same zero-bias potential energy VO and (b) layers 2001 - 200M are chosen to have, alternately, low potential energy, such as, for example, Vl, and high potential energy, such as, for example, V2. As a result of these specific choices and in accordance with the mapping of eqn. (2) above where ne is shown to be proportional to the square root of (E - V), i.e., the electron kinetic energy, layers 2001 - 200M have, 2no.?1 .~4 alternatively, high refractive index and low refractive index.
In accordance with the embodiment of present invention shown in FIG. 5, when predetermined bias potential Vbias is applied to filter/emitter 100, layers 2001 - 200Rl form a reflector R1 where each layer of reflector Rl is a quarter of an electron wavelength in thickness as measured in that layer at the passband energy Ep, layer 200R1+1 is one half of an electron wavelength in thickness as measured in that layer at the passband energy Ep, and layers 200R1+2 - 200M form a reflector R2 where each layer of reflector R2 is a quarter of an electron wavelength in thickness as measured in that layer at the passband energy Ep. Further, an electron wavelength of a layer is determined in accordance with eqn. (1) to be given by:

electron wavelength = 2 ~/[2m (E - V)]1/2 (B3) The embodiment of the present invention shown in FIG. -5 is formed from a material system comprised of a continuous set of alloys of the type F1_XGxH. In general, such a material system will have a restricted range of usable compositions, such as the range of usable compositions denoted by the range o< x < xmax, because there may be a transition at xmax from a direct to an indirect energy gap like that which occurs in the Ga1_*1 *s material system at x = 0.45.
In addition, in the embodiment of the present invention shown in FIG. 5, layers 11001 and 11002 which surround filter/emitter 100 are formed from the same material system and comprise an alloy wherein x = xO. As is well known to those of ordinary skill in the art, the electron potential energy in a layer of material comprised of Fl_XGxH
may be-given by:
Vj = delEc = Axj (B4) - 200~134 ~ 28 where delEc is the change in the energy of the conduction band edge in the material and A is a constant. In accordance with eqn. (B4), the range of potential energies for the given material system in fabricating embodiments of the present invention is given by 0 < V < Vmax. This usable range of potential energies is shown in FIG. 5 by the spread in potential energy between dotted line 150, indicative of 0 electron potential energy for the given material system, and dotted line 170, indicative of Vmax electron potential energy for the given material system.
Still further, in order for filter/emitter 100 to be physically realizable, the thickness dj of each of layers 2001 - 200M must be an integer multiple, pj, of the monolayer thickness rj for the material composition of that layer.
The inventive method for providing embodiments of the present invention, comprises, as a first step, selecting a suitable unbiased superlattice electron wave interference filter like that disclosed above in the section discussing unbiased devices. Specifically, a nine layer filter, i.e., M = 9, was disclosed wherein the nine layers were comprised of 72 monolayers of semiconductor material. We will now describe how an embodiment of the present invention is designed in accordance with the remaining steps of the inventive method so that specific material compositions and layer thicknesses are determined. In particular, the disclosed embodiment of inventive filter/emitter 100 will filter electrons having an input kinetic energy of 0.10 eV, i.e., KEin = 0.10 eV, and emi~ them with a kinetic energy cf 0-20 eV, i.e., KEoUt = 0.20 eV. In this case, since Vbias equals the difference between the output and input kinetic energy, i.e., output kinetic energy (KE)oUt is equal to Vbias + (KE)in, Vbia5 will equal 0.10 eV.
In the notation of optical thin film design, H
indicates a quarter-wavelength thickness layer of high index (as measured in the medium), and L indicates a quarter-2003~ ~4 wavelength thickness layer of low index (as measured in themedium). Thus, the notation HH signifies a half-wavelength thickness layer of high index. The disclosed embodiment of inventive filter/emitter 100 comprises nine layers wherein:
(l) layer 1 is a high index, quarter-wavelength layer (H);
(2) layer 2 is a low index, quarter-wavelength layer (L); (3) layer 3 is a high index, quarter-wavelength layer (H); (4) layer 4 is a low index, quarter-wavelength layer (L); (5) layer 5 is a high index, half-wavelength layer (HH); (6) layer 6 is a low index, quarter-wavelength layer (L); (7) layer 7 is a high index, quarter-wavelength layer (H); (8) layer 8 is a low index, quarter-wavelength layer (L); and (9) layer 9 is a high index, quarter-wavelength layer (H).
We will now describe the inventive design method.
First, we will describe how the thickness of a semiconductor layer which is a multiple of a quarter of an electron wavelength at a given electron pass energy Ep is determined.
For the jth layer of filter/emitter 100 to be a quarter of an electron wavelength in thickness at the pass energy Ep, the phase difference of the electron wave between the input boundary to the jth layer, i.e., at Zj-l in FIG. 5, and the output boundary to the jth layer, at zj in FIG. 5, must be an odd multiple of ll/2 given by (2qj - 1)ll/2. This condition is written as follows:

~ kj dz = ~ {2m*jtEp - Vj(z)]}l/2/~dz Zj_l Zj_l = (2qj - 1) ~/2 (B5) The potential energy Vj(z) in the jth layer is given by:
j( ) VbiaS(1-z/L) + Vj (B6) where L is the total length of superlattice 100 and qj is a positive integer.

200~1a4 The pass energy for filter/emitter 100 is given as:
Ep = Vbias + VO + (KE)in (B7) where (KE)in is the pass kinetic energy in input layer 11001.
The effective mass of the jth layer is given by:
m j = (B + Cxj)mO (B8) where B and C are constants and mO is the free electron mass.
Using Vj = Axj one obtains the following "quarter-wavelength" condition:
~2L[2mO(B + Cxj)]l/2/3~vbias~*
{[V +(KE)in-Axj+vbiaszi/L] / - [VO+(KE)in Ax; bias j-3/2) = (2qj - 1) ~/2 (B9) Eqn. (B9) is solved to determine the composition Xj of the jth layer.
Let:
(1) i be an index which counts the number of monolayers of material which comprise inventive filter/emitter 100;
(2) ij denote the number of the rightmost monolayer in the jth layer of inventive filter/emitter 100; and (3) iM denote the total number of monolayers in inventive filter/emitter 100.
Using this notation:
(1) the thickness of the jth layer is given by dj = pjri where pj = ij - ij-l is the number of monolayers in the jt layer and rj is the thickness of a monolayer of material composition x; for the jth layer;
(2) the total thickness of inventive filter/emitter 100 is given by L = sum of pjri which is equal to iMr for a material 2003~34 system having rj be the same value, r, for all layers;
(3) the distance along inventive filter/emitter 100 from one end to the beginning of the jth layer is given by j-l d Zj_1 (ij_l/iM)*L when all r are the same; and (4) the distance along inventive filter/emitter 100 from one end to the end of the jth layer is given by Zj and z; = (ij/iM)*L when all rj are the same.
In terms of this notation, the inventive method comprises the following steps:
(1) for the first layer, set io equal to 0 and set j = 0.
(2) increment the value of ij for the next value of j by 1, for example, i1=1. For quarter-wavelength layers, set qj equal to 1 and, for half-wavelength layers, use ¦¦
instead of ~ /2, where (2qj - 1) is the number of quarter wavelengths for the jth layer.
(3) solve eqn. (B9) for xj using j, the previously determined value of ij_1, and the preset value of ij, where x; is the composition of the jth layer. For a high electron refractive index layer, choose the positive real value of x;
which is closest to zero and, for a low electron refractive index layer, choose the resulting real value of x; which is closest to, but less than the value where a transition from a direct band gap to an indirect band gap occurs, for example, this value is 0.45 for the GaAlAs material system. If there are more layers to do, go back to step 2, otherwise go on to step 4.
(4) If the total number of monolayers that have been determined after the last step has been completed are more than or less than the initial estimate of the number of 2003i:34 monolayers iM~ then one must revise the initial estimate and go back to step one to try again.
The above-described process is repeated until the optimum thickness, corresponding to the value of xM closest to zero, of the last layer produces a total thickness for inventive filter/emitter 100 which is in self-consistent agreement with the initial estimate of iM used in the design.
We have applied the inventive method to design inventive filter/emitter 100. As set forth above, in this embodiment, layers 2001 to 2009 and surrounding layers 11001 and 11002 are comprised of materials from the Ga1_xAlxAs material system. This is an advantageous material system because all compositions are lattice matched for these alloys and because, for growth along the [100] direction, the monolayer thickness is the same, i.e., rj = r = 2.82665 A.
Further, a composition in this material system is a direct gap semiconductor for x less than or equal to 0.45 and, as a result, this represents the usable composition range. Still further, for Ga1_xAlxAs, A = 0.77314 eV, B = 0.067 and C = 0.083. Yet still further, in this embodiment we will use surrounding layers 11001 and 1l002 which are comprised of the same composition, GaO 55Alo 45As, having xO = 0.45.
The design for the nine-layer embodiment of filter/emitter 100 in the Ga1_xAlxAs system having input kinetic energy of 0.10 eV and output kinetic energy of 0.20 eV, which embodiment might be useful as an emitter in a high speed ballistic transistor, is set forth in TABLE B-A.
The effective masses used in the design of filter/emitter 100 were taken from prior art literature to be m (GaAs)=0.067mO
and m (GaO 55Alo 45As)=0.10435mO, where mO is the free electron mass. The conduction band edge energies used were V(GaAs)= o.0000 eV and V(GaO 55Al0.45AS) = 0-3479 eV- In this design, the total thickness of inventive filter/emitter 100 is 71 monolayers, which thickness corresponds to a length L = 20.0692 nm. This illustrates the fact that the length of 200~ 4 such an embodiment of inventive filter/emitter 100 is short enough so that it may be fabricated from thin layers of ballistic semiconductor materials.
As a check on the methodology presented above, we have computed the electron current transmittance for the biased superlattice design described in TABLE B-A. Since the bias provides a linear potential decrease, the electron wavefunction in any layer of filter/emitter 100 may be expressed as a linear combination of Airy functions Ai(~) and complementary Airy functions Bi()o), where a new variable is defined within the jth layer as follows:
= (2m jVbias/~2L)1/ {Z+(E-Vbias-Vj)L/vbias~ (B10) For the stack of M layers shown in FIG. 5, the electron transmittance Te is given by the following factor multiplied by the square of the electron amplitude transmittance:
[(E - VO - Vbias)/M o]1/2/[(E - Vo)/M M+1]1/2 (B11) Te is shown in FIG. 6 for the biased superlattice design described in TABLE B-A. At the design bias of Vbias =
0.10 eV, i.e., curve 300, the device emits (KE)oUt = 0.20 eV
electrons into output layer 11002. The full-width-at-half-maximum (FWHM) is 30.7 meV or 15.35% of the center energy.
Further, the output kinetic energy from inventive filter/emitter 100 is continuously tunable, i.e., the peak of the curve can be shifted and the curve maintains a shape which provides for a filtering function. In particular, the peak of the output kinetic energy is shifted by changing the bias potential energy Vbias applied to inventive filter/emitter 100. Since the HH resonant layer is at the center of the device, as measured in electron optical path length, the shift in the peak of the output kinetic energy is equal to one-half of the change in the bias potential energy, i.e., as the bias potential energy is changed by 50 meV, the peak of the output kinetic energy is shifted by 25 meV.
Thus, as shown in FIG. 6, for a change in Vbias of + 50 meV, 200.~1~.'34 _ curves 301 and 302 show that the peak of the output kinetic energy is shifted by + 2S meV. Note however, that the transmittance for the different curves is different. This property of continuous tunability provides a substantial advantage for embodiments of the inventive filter/emitter, aside from flexibility, in that the bias potential energy may be varied to tune an embodiment which is fabricated with output characteristics that deviate from strict conformance with a predetermined design.
Certain important characteristics of all-dielectric optical interference filters which were discussed above in the section concerning unbaised devices also apply to the inventive solid state electron wave filter/emitters, namely:
(CH2); (CH3); (CH4); (CH5); (CH6); (CH7); (CH8); (CH9); and (CH10).
It should be clear to those of ordinary skill in the art that the inventive superlattice devices may be comprised of layers which have a multiplicity of differing material compositions. Further, it is well known to those of ordinary skill in the art as to how a bias potential Vbias may be applied to inventive filter/emitter 100 by, for example, applying electrodes to layers 2001 and 200M of FIG.

5 and by applying a source across these electrodes.
Clearly, those skilled in the art recognize that further embodiments of the present invention may be made without departing from its teachings. For example, it is within the spirit of the present invention to provide a wide variety of hole wave devices as well as electron wave devices. In addition, it is within the spirit of the present invention that a wide variety of electron or hole wave filter/emitter devices can be fabricated. Further in addition, such inventive devices may be used to provide narrowband semiconductor superlattice interference filter/emitters for use as hot electron emitters for an entire class of devices such as electroluminescent devices, 2nQ.~

_ 35 photodetectors, and ballistic transistors and so forth.
Still further in addition, such inventive devices may be used to aid in controlling, shaping and filtering freespace electron beams to provide electron spectrometers, electron lithography and electron diffraction analysis of crystals.
For both the unbiased devices and the biased devices, it is important to note that the extent to which practical embodiments of the inventive devices conform with the designs described herein depends on the amount of ballistic motion which occurs in the materials out of which they are fabricated. This means that the behavior of the inventive devices will more closely resemble the desired and designed characteristics if electron transport within the materials is substantially ballistic. However, it is also important to note that the inventive devices will also perform in accordance with the designed characteristics, albeit in a degraded fashion, if electron transport is not substantially ballistic, i.e., their performance will "gracefully" degrade. Nevertheless, present fabrication techniques in molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD), enable those of ordinary skill in the art to grow multilayered superlattice structures with precise monolayer compositional control and with materials which provide substantially ballistic electron transport. In addition, doping of semiconductors is not important for the embodiments of the present invention, however, it is preferred to exclude doping within the active region of the device in order to avoid scattering within the materials. This provides a further advantage for the inventive devices because the absence of doping makes them easier to fabricate.
Although there is no prohibition in principle against the use of indirect bandgap materials, they cannot be used where the transition between a direct and an indirect bandgap material or between two indirect bandgap materials 2n~ 4 requires a change in momentum. This is because we are dealing with wave effects that occur in substantially collisionless motion.
Note that, like their thin-film optical counterparts, semiconductor superlattice interference filters for the unbiased case and filter/emitters for the biased case will be relatively insensitive to variations about the design composition values. Further: (1) although semiconductor materials (a) may have nonparabolic bandstructure in terms of E vs. k, i.e., energy vs. momentum, and (b) may have band structures which vary with a particular direction of electron wave propagation in the material, i.e., anisotropic energy band structure, these effects, if present, may be incorporated into the inventive design method by using an energy dependent, anisotropic effective mass m . Thus, even though an allowed wavevector surface is no longer spherical in the presence of anisotropy, all of the inventive design methods set forth herein still apply, provided the energy dependent anisotropic effective mass is used in the analysis.
Lastly, for both the unbaised devices and the biased devices, it should be clear to those of ordinary skill in the art that appropriate solid state materials for use in fabricating embodiments of the present invention include semiconductor materials such as, without limitation, binary, ternary and quaternary compositions of, among others, III-V
elements and II-VI elements.
WAVEGUIDES
FIG. 7 shows, in pictorial form, the energy level diagram and the structure of asymmetric, quantum well slab waveguide 2100. We will use the following notation in describing this embodiment of the present invention:

2no~ 4 (a) layer 2200 will be referred to as substrate layer s, layer 2201 will be referred to as film layer f, and layer 2202 will be referred to as cover layer c;
(b) the direction perpendicular to the surfaces of waveguide layers 2200-2202 denoted by arrow 2100 will be referred to as xw;
(c) the electron potential energy at the bottom of the quantum well in layer 2201, i.e., film layer f, will be referred to as Vf;
(d) the electron potential energy barrier heights associated with layer 2200, i.e., substrate layer s, and layer 2202, i.e., cover layer c, will be referred to as VS and Vc, respectively;
(e) layers 2200-2202 are comprised of materials from a material system of the type Fl_XGxH, and, as a result, we will refer to the compositions of layers 2200-2202, i.e., substrate layer s, film layer f, and cover layer c, as x5~ Xf, and xc~
respectively;
(f) the direction of guided mode propagation denoted by arrow 2110 will be referred to as Zw;
(g) the thickness of waveguide layer 2201, i.e., film layer f, will be referred to as d;
(h) the angle of incidence of the two plane wave components that constitute an electron guided wave will be referred to as the zig-zag angle O;
(i) the magnitude of the electron wavevector in any of layers 2200-2202 is given by ki = [2m*i(E - vi)]l/2/~, where i = s, f, c for substrate layer s, i.e., layer 2200, film 2no~ 4 layer f, i.e., layer 2201, and cover layer c, i.e., layer 2202, respectively, and where m i is the electron effective mass, Vi is the electron potential energy, and E is the total electron energy.
More specifically, in the following description, layers 2200-2202 are comprised of materials from the Gal_xAlxAs material system and, as a result, the electron potential energies in layers 2200-2202 of waveguide 2100, i.e., Vs~ Vf, and Vc, respectively, are given by the conduction band edge as:
Vi = Axi i = s, f, c (S3) Further, the electron effective masses in layers 2200-2202 of waveguide 2100 are given by:
m = (B + Cxi)mO i = s, f, c (S4) where mO is the free electron mass.
Before describing the inventive method for designing specific embodiments of the inventive slab waveguide in detail, we will qualitatively describe the manner in which inventive slab waveguide 2100 operates. This will better enable one to understand the inventive method.
Further, one can better understand the manner in which inventive slab waveguide 2100 operates by understanding the concept of critical angle as it applies to the present invention.
In particular, an equivalent of Snell's law for electron waves, as it relates to inventive slab waveguide 2100, is developed by requiring that the component of the electron wavevector which is parallel to a boundary between two layers be the same before and after reflection and refraction, i.e., by requiring that the phase of the transmitted and reflected electron waves along a boundary between two layers be identical to the phase of the incident electron wave. In accordance with this, the onset of total internal reflection occurs when the angle of incidence, i.e., Z~)0:~34 the zig-zag angle defined above, is equal to the critical angle. The critical angle is given by:
e~ if = sin~1 {[m*i(E - vi)]/[m*f(E - Vf)]}1/2 (S5) for Vi < E < Eif where: (a) i = s for the critical angle at the boundary between layers 2200 and 2201, i.e., the boundary between substrate layer s and film layer f;
(b) i = c for the critical angle at the boundary between layers 2201 and 2202, i.e., the boundary between film layer f and cover layer c; and (c) Eif = (m iVi ~ m fVf)/(m i ~ m*f) This is interpreted physically as follows. An electron wave which is incident upon a boundary at an angle which is greater than e~ if will be totally reflected if the layer on the other side of the boundary is infinitely thick.
Thus, at steady state, all of the incident electron current from the film layer which is incident, for example, on an infinitely thick substrate layer or on an infinitely thick cover layer will be reflected back into the film layer. It is interesting to note that if the kinetic energy of an electron wave is less than or equal to 0, i.e., (E - Vi) < 0, then total internal reflection can occur for any angle of incidence, including normal incidence. This is different from the case of electromagnetic waves where total internal reflection can never occur at normal incidence due to the non-zero value of the refractive index.
FIG. 8 shows, in pictorial form, a plot of electron propagation constant versus total el~ctron energy when substrate layer 2200 of slab waveguide 2100 is comprised of GaO 85Alo 15As, film layer 2201 is comprised of GaAs, and cover layer 2202 is comprised of GaO 70Alo 30As. For an infinite medium, the electron propagation constant is defined as follows: bi = [2m*i(E - vi)]1/2/~, where i = s, f, c for 2(~0.~ 4 substrate layer s, i.e., layer 2200, film layer f, i.e., layer 2201, and cover layer c, i.e., layer 2202, respectively. As shown in FIG. 8: (a) curve 500 is a plot of b5 for substrate layer s, i.e., layer 2200; (b) curve 501 is a plot of bf for film layer f, i.e., layer 2201; and (c) curve 502 is a plot of bc for cover layer c, i.e., layer 2202. The interpretation of the information provided by curves 500-502 shown in FIG. 8 is as follows. For a given total electron energy E, the propagation constant of a guided mode can be no larger than bf. As a result of this, region 600 to the left of curve 501 corresponds to evanescent or non-physical modes and the allowed guided modes for this electron waveguide must lie to the right of curve 501.
However, an allowed guided mode must satisfy the condition that its zig-zag angle must be greater than the critical angle e ' cf at the cover-film boundary and the critical angle e'sf at the substrate-film boundary, i.e., the range of zig-zag angle and total energies in FIG. 8 must satisfy the condition that e > max [e'cf~ e'sf] As a result of this, allowed guided modes must lie in region 601 to the left of curve 500.
Next we now will qualitatively discuss cutoff phenomena as they relate to inventive slab waveguide 2100.
An electron guided wave can become cutoff by decreasing the electron energy and we will refer to the energy at which this cutoff occurs as the lower-energy cutoff. The zig-zag angle of the plane wave component of the electron wave decreases with decreasing energy and lower-energy cutoff occurs when zig-zag angle ~ = 0. The propagation constant bv of the vth guided mode, where v is an integer starting at 0, is given by:
bv = t2m f(E - V)/~2]1/2sine (S6) As a result, lower-energy cutoff occurs when bv = - At this point, the wavefunction is sinusoidal in film layer f, i.e., layer 2201, and exponentially decaying in substrate layer s, 200~ 4 i.e., layer 2200, and in cover layer c, i.e., layer 2202. In this sense, lower-energy cutoff is analogous to the cutoff of an electromagnetic guided mode in a hollow metallic waveguide with finite conductivity walls where the plane wave components of the electromagnetic guided wave are reflecting back and forth at normal incidence to the waveguide boundaries.
As the electron energy of a guided mode is increased, an upper-energy cutoff will also occur. The upper-energy cutoff can be of three types: (1) cutoff to a substrate mode in region 602 of FIG. 8 which is like the cutoff in an electromagnetic asymmetric dielectric waveguide where the substrate index of refraction is higher than the cover index of refraction; (2) cutoff to a radiation mode in region 603 of FIG. 8 which is like the cutoff in an electromagnetic symmetric dielectric waveguide having equal substrate and cover indices of refraction; and (3) cutoff to a cover mode which is like cutoff in an electromagnetic asymmetric dielectric waveguide with the cover index of refraction higher than the substrate index of refraction.
The type of upper-energy cutoff which occurs in inventive slab waveguide 2100 depends upon the intersection of propagation constants bS, bf, and bc. Specifically, the intersection of bf and b5 occurs at energy Esf which is given by:
Esf = (m 5Vs ~ m fVf)/(m s ~ m*f) (S7) At this energy, the electron wave phase refractive indices for film layer f, i.e., layer 2201, and substrate layer s, i.e., layer 2200, are equal and, when this energy is reached, inventive slab waveguide 2100 can no longer guide an electron wave, even if it is at grazing incidence along the walls of waveguide 2100. This energy, Esf~ is equivalent to substrate-film critical angle e'sf = 90 200:~i34 Similarly, the intersection of bf and bc occurs at energy Ecf which is given by:
ECf = (m cVc ~ m fVf)/(m c ~ m*f) (S8) At this energy, the electron wave phase refractive indices for film layer f, i.e., layer 2201, and cover layer c, i.e., layer 2202, are equal. This energy, Ecf~ is equivalent to cover-film critical angle e'cf = 90 Similarly, the intersection of b5 and bc occurs at an energy ECs which is given by:
Ecs = (m cVc ~ m sVs)/(m c ~ m*5) (S9) At this energy, the electron wave phase refractive indices for substrate layer s, i.e., layer 2200, and cover layer c, i.e., layer 2202, are equal.
In general, the type of upper-energy cutoff that occurs depends on the material parameters. In particular, in the embodiment depicted in FIG. 8, upper-energy cutoff will be to a substrate mode because b5, plotted as curve 500, occurs at a lower energy in general that does bc, plotted as curve 502.
In summary, as the electron energy increases for an electron guided mode, zig-zag angle e also increases.
Further, when zig-zag angle O reaches critical angle e~sf~
the electron guided wave starts to refract into substrate layer s, i.e., layer 2200, rather than exponentially decaying therein. Then, for electron energies greater than the energy at which zig-zag angle O equals critical angle e~sf~ the electron wave propagates in substrate layer s as well as in film layer f. This condition will be referred to as a substrate mode. Finally, as the electron energy is further increased so that zig-zag angle O reaches critical angle e ' cf~ the electron wave starts to refract into cover layer c, i.e., layer 2202, as well as into substrate layer s. At this point the electron wave is propagating in all three layers, i.e., layers 2200-2202, and will be referred to as a radiation mode. Further note, that for a different set of material parameters, as the electron energy is increased, it is possible for the electron wave to be refracted into cover layer c and will be referred to as a cover mode.
The following describes the inventive method which is used to determine the thicknesses and compositions for specific embodiments of inventive slab waveguide 2100.
As described above, the magnitude of the electron wavevector and the electron wave phase refractive index ne(amplitude) of an electron wave in any of layers 2200-2202 is given by eqns. (1) and (2). Further, the wavefunction for a two-dimensional (XW, Zw) quantum well guided electron wave has a sinusoidal dependence in the Zw direction and can be expressed as:
Uv(Xw~Zw) = Uv(Xw)exp(jbvzw) (S10) where bv is the guided mode propagation constant.
Using eqn. (S10), the Schroedinger time-independent wave equation becomes:
d2Uv(xw)/dxw2 + (2m*/h2)[Ev-V(xw)]-bv2}Uv(xw) = (S11) Thus, for a guided mode, the wavefunction amplitude in substrate layer s, i.e., layer 2200, is given by:
Uvs(xw) = AseXP(gsXw) (S12) the wavefunction amplitude in film layer f, i.e., layer 2201, is given by:
Uvf(Xw) = Aflexp(jkfxw) + Af2exp( j f w) (S13) and the wavefunction amplitude in cover layer c, i.e., layer 2202 is given by:
Uvc(Xw) = AceXP(-gc(xw - d)) (S14) where:
gC2 = bV2 - [(2m c/~2)(EV ~ Vc)]
kf2 = [(2m*f/~2)(Ev - Vf)] - bV2 (S15) gS2 = bV2 - [(2m s/~2)(EV ~ Vs)]
The dispersion equation for guided modes in film layer f, i.e., layer 2201, is determined by using the boundary condition that U and (1/m*)(dU/dx) are continuous 200:~i3~

across the cover-film and substrate-film layer boundaries.
The dispersion equation for guided modes in film layer f is:
kfd - tan l[(gS/m*s)/(kf/m f)]

- tan~l[(gc/m*c)/(kf/m f)] = v ~ (S16) where v is the integer mode number. Further, we will denote the guided electron waves as Mv.
Lower-Ener~y Cutoff:
As discussed above, lower-energy cutoff for guided modes Mv can occur as the electron energy is decreased and guided wave propagation constant bv goes to zero so that the mode is no longer propagating. FIG. 8 shows that this can occur only for an electron energy below the lower barrier energy of slab waveguide 2100, i.e., when E < Vs~
The electron energy at which lower-energy cutoff occurs is designated as ELCo and the condition for lower-energy cutoff is determined by substituting bv = into dispersion eqn. (S16). As a result, the lower-energy sutoff condition is determined by solving the following transcendental equation for ELCo corresponding to the Mv mode:
[2m f(ELCo ~ Vf)]l/2d/~
- tan l[m f(Vs - ELCo)/m*s(ELco Vf)]

- tan l[m f(Vc ~ ELco)/m c(ELc f)] ~ (S17) UpPer-Energy Cutoff:
As discussed above, upper-energy cutoff for guided electron waves Mv can occur as the electron energy is increased and total internal reflection no longer occurs, for example, at the substrate-film layer boundary. Thus, as the electron energy is increased through an upper-energy cutoff, the electron wave is refracted into the substrate. That is, as the electron energy is increased through the upper-energy cutoff, the electron wave function amplitude in the substrate changes from being evanescent, i.e., exponentially decaying, Z00~ 4 -to propagating, i.e., sinusoidal. This can only occur for an electron energy above the lower barrier energy of slab waveguide 2100, i.e., when E > Vs~ As a result, the mode "leaks" into the substrate.
The electron energy at which upper-energy cutoff occurs is designated as EUco and the condition for upper-energy cutoff is determined by substituting gs = into dispersion eqn. (S16). As a result, the upper-energy cutoff condition for cutoff to substrate modes is determined by solving the following transcendental equation for EUco corresponding to the Mv mode:
~2tm SVs ~ m fVf - (m s ~ m f)Euco])1/2d/~
- tan-l{A/s~l/2 = v ~ (S18) where: A = [m*cVc ~ m sVs ~ (m c ~ m s)Euco] f B = ~m*5Vs - m fVf - (m s ~ m f)EucO]m c EnergY of the First APPearance of Modes:
For a given set of layer compositions and potential energies, as waveguide thickness d is increased, i.e., thickness d of layer f or layer 2201 is increased, a guided mode Mv first starts to propagate at an energy E - Vs~ This energy corresponds to the highest possible value of cutoff energy for lower-energy cutoff as well as the lowest possible value of cutoff energy for higher-energy cutoff.
Substituting E = Vs into dispersion eqn. (S16) gives the thickness d at which mode Mv first starts propagating as:
d = (~/2m f(Vs ~ Vf)]1/2~

*{tan~1[m*f(Vc - Vs)/m*c(Vs - Vf)]l/2 + v~} (Sl9) Also, the range of thicknesses that will produce a waveguide that supports only modes including the vth mode is given by:

1 [K2 + v ~]< d < Kl * [K2 + (v + 1) ~] (S20) where: Kl = ~/[2m f(Vs ~ Vf)] /
K2 = tan~1[m f(Vc ~ VS)/m c(Vs f)]

2003~34 Thus, one can use eqn. (S20) with v = 0 to determine the range of thicknesses for layer 2201 so that only the lowest mode, i.e., Mo~ is guided. As one can readily appreciate from this, as with electromagnetic asymmetric dielectric slab waveguides, there is a minimum thickness required for any modes to propagate.
In FIG. 8, curves 2300-2302 are mode dispersion curves, i.e., plots of propagation constant bo for the lowest electron guided mode Mo as a function of total electron energy for a specific embodiment of inventive slab waveguide 2100. Specifically: (a) layer 2200, i.e., substrate layer s, is comprised of GaO 85Alo 15As, i.e., XS = 0.15; (b) layer 2201, i.e., film layer f, is comprised of GaAs, i.e., and Xf = 0; and (c) layer 2202, i.e., cover layer c, is comprised of GaO 70Alo 30As, i.e., XC = 0.30. We have used eqn. (S3) with A = 0.7731 eV and we have taken the conduction band discontinuity to be approximately 60% of the energy gap change to determine Vs = 0.115971 eV for layer 2200, Vf = O.OOOo eV for layer 2201, and Vc = 0.231942 eV for layer 2202. In addition, we have used eqn. (S4) with B = 0.067, and C = 0.083 to determine m*5 = 0.07945mO for layer 2200, m f = 0.067mO for layer 2201, and m*c = O.O919mO for layer 2202. In addition, for this embodiment, we have taken layer growth to be along the [100] direction and, as result, each monolayer of material for waveguide 2100 has a thickness of 0.28267 nm.
Using eqn. (S19), we find that the fundamental mode Mo starts propagating when film layer f has a thickness d of 6 monolayers. Further, again using eqn. (Sl9), we find that the next mode, M1 starts propagating at a thickness of 31 monolayers. As a result, for this embodiment, slab waveguide 2100 acts as a single mode waveguide for GaAs layer 2201 thicknesses of from 5 to 30 monolayers.
Curves 2300-2302 shown in FIG. 8 are solutions of dispersion eqn. (S16) for bo as a function of total electron 2()03134 energy for various thicknesses d of film layer f.
Specifically: (a) curve 2300 corresponds to a film layer f, i.e., layer 2201, thickness of 10 monolayers of GaAs, i.e., d = 2.82665 nm; (b) curve 2301 corresponds to a thickness of 20 monolayers of GaAs, i.e., d = 5.6533 nm; and (c) curve 2302 corresponds to a thickness of 30 monolayers of GaAs, i.e., d = 8.47995 nm.
As one can readily appreciate from FIG. 8, as the thickness of film layer f, i.e., layer 2201, is increased, mode dispersion curves 2300-2302 move to the left and upward.
Further, as the thickness d of film layer f increases, the lower-energy cutoff decreases, i.e., curve 2301 crosses the energy axis at lower energy than curve 2300 does, and the upper-energy cutoff increases, i.e, curve 2301 intercepts curve 500 at higher energy than curve 2300 does. Note that even at a thickness of 10 monolayers, a guided mode such as Mo can propagate at energies above both potential barriers, i.e., E > Vs and E > Vc. This is seen by the fact that curve 2300 intersects curve 500 at a point which is at an energy above both Vc and Vs~
The energy difference,~ E, between upper-cutoff energy EUco~ i.e., cutoff to a substrate mode, which is determined from eqn. (S18) and corresponds to the intersection of an Mo curve 2300-2302 with curve 500 of FIG.
8, and lower-cutoff energy ELCo~ i.e., cutoff at bv = ~
which is determined from eqn. (S17) and corresponds to the intersection of an Mo curve 2300-2302 with the energy axis, is set forth for the thicknesses of curves 2300-2302 in Table S-I.
Symmetric Wavequides:
In an embodiment of the inventive slab waveguide 2100 which is symmetric, i.e., Vc = Vs~ substrate and cover dispersion curves bS and bc coincide, where substrate dispersion curve bS = [2ms*(E-Vs)]1/2/~ and cover dispersion curve bc = [2mC (E-VC)]l/2/~. In this case, lower-energy cutoff again occurs as the electron energy is decreased and propagation constant bv goes to zero, i.e., bv = - This can only occur for an electron energy below the lower barrier energy, i.e., E < Vs = Vc. When bv = , zig-zag angle e also equals 0 and the plane wave components of the guided wave are reflected back and forth at normal incidence to the waveguide boundaries.
For a symmetric waveguide, upper-energy cutoff occurs as the energy is increased and the guided mode becomes a radiation mode. At the upper-energy cutoff, total internal reflection occurs neither at the substrate-film boundary nor at the cover-film boundary. The electron wave is then refracted both into substrate layer s, i.e., layer 2200, and cover layer c, i.e., layer 2202. As the electron energy is increased through upper-energy cutoff, the electron wave function amplitude in substrate layer s, i.e., layer 2200, and cover layer c, i.e., layer 2202, changes from being evanescent to propagating. This can only occur for an electron energy above the barrier energy, i.e., E > Vs = Vc-When g5 = gc = , the mode leaks into substrate layer s,i.e., layer 2200, and cover layer c, i.e., layer 2202. This is analogous to the cutoff of an electromagnetic guided mode in a symmetric dielectric slab waveguide. This type of cutoff occurs when the zig-zag angle becomes simultaneously equal to the substrate-film critical angle and the cover-film critical angle.
The first appearance of electron guided wave Mv occurs when the electron energy E = Vs = Vc. As a result, electron guided wave Mv first starts propagating as the thickness d of film layer f, i.e., layer 2201, is increased to the value:
d = v~ ~/[2m*f(Vs ~ Vf)]l/2 (S21) 200:~134 _- 49 Also, from eqn. (S20), the range of thicknesses that will produce a waveguide that supports only the lowest order, i.e., v = 0, mode Mol is obtained from:

0 < d < ~ ~/[2m f(Vs ~ Vf)]l/2 (S22) As one can appreciate from eqn. (S22), symmetric electron slab waveguides are similar to electromagnetic symmetric slab waveguides in that there is no minimum thickness required for the lowest-order mode to propagate, i.e., any thickness will support the Mo mode. However, for very thin electron slab waveguides, the exponentially decaying tails of the wavefunction may extend very far into the substrate and cover layers.
FIG. 9 shows, in pictorial form, wavefunction Uv for the Mo mode for a 10 monolayer thick GaAs film layer f, i.e., layer 2201, for various electron energies: (a) curve 401 corresponds to electron energy El = ELCo with b = 0 cutoff; (b) corresponds to electron energy Vs < E2 ~ Vc; (c) corresponds to electron energy Vc < E3 < EUco; (d) corresponds to electron energy E4 = EUco; and (e) corresponds to electron energy E5 > EUco~ Curves 401-404 illustrate guided mode behavior in those ranges of electron energy and curve 405 shows the wavefunction for an energy above upper-cutoff energy to illustrate substrate mode behavior.
Clearly, those skilled in the art recognize that further embodiments of the present invention may be made without departing from its teachings. For example, it is within the spirit of the present invention to provide a hole slab waveguide as well an electron slab waveguide.
In terms of nomenclature, it should be clear to those of ordinary skill in the art that references to electron energies being above the potential barriers, correspond to energies, as shown in FIG. 7, which are above the conduction band. Further, it should also be clear to those of ordinary skill in the art that similar references _ 50 for holes correspond to energies which are below the valence band.
Further, it well known to those of ordinary skill in the art as to how electrons and/or holes may be injected into the film layer of a slab waveguide.
Still further, it should be clear to those of ordinary skill in the art that embodiments of the present invention may be fabricated wherein the film layer is comprised of a substantially ballistic material whereas the substrate layer and/or the cover layer are not so comprised.
However, in such embodiments it would be advantageous for the doping of the substrate layer and/or the cover layer to be small enough so that excessive loss in these layers is not caused thereby.
Yet still further, it should be clear to those of ordinary skill in the art that the thickness of the substrate layer and the thickness of the cover layer may have substantially any value. In practice, however, the thicknesses of these layers should be large enough to support the exponential tails of guided waves in the film layer.
Such a thicknesses are typically much less than the thickness of the film layer and, in general, will be a few monolayers of the material of which the substrate layer and/or the cover layer is comprised. In practice, the thickness of the substrate layer is of no concern because the substrate layer is typically much thicker than layers which are grown or deposited thereon. Physically, these requirements may be understood as defining a requirement that the guided wave in the film layer not "sense" the presence of a boundary at, for example, the top of the cover layer.
Lastly, it should be clear to those of ordinary skill in the art that h is Planck's constant, that h is Planck constant divided by 2~ , and that appropriate solid state materials for use in fabricating embodiments of the present invention include semiconductor materials such as, 20031~4 without limitation, binary, ternary and quaternary compositions of, among others, III-V elements and II-VI
elements.
TABLE I
Design Parameters for a Narrow-Band Optical Interference Filter of the 1.0 HL HH LHLHL HH L'H 1.0 Type Optical Thin Film Filter (Pass Wavelength = 1.00 um) Refractive Thickness Index (A) Input Region 1.00 ...
Region 1 (H) 4.00 625.000 Region 2 (L) 1.35 1851.852 Region 3 (L') 1.83 1366.120 Output Region 1.00 ...
TABLE II
Design Parameters for a Narrow-Band Electron Superlattice Interference Filter Which Corresponds to an Optical Interference Filter of the 1.0 HL HH LHLHL HH L'H 1.0 Type 20 Electron Superlattice Filter (Pass Wavelength = 100 A) Kinetic Thickness Energy (eV) (A) Input Region 0.015037 ...
Region 1 (H) 0.240592 5.25000 Region 2 (L) 0.027405 18.51852 Region 3 (L') 0.050357 13.66120 Output Region 0.015037 ...

Z0031:~4 TABLE III
Calculated compositions, values of x in Gal_~Al~s, that produce one quarter-wavelength layers for ~he kinetic energies and number of monolayers indicated. In all cases, the surrounding material is GaO 55Alo 45As and the monolayers are crystalline (100) planes and only compositions where x is less than or equal to 0.45 are used.

Pass Number of Monolayers 10Kinetic Energy (eV) 6 7 8 9 10 0.140 0.00150.2902 0.3923 0.4513 0.4898 0.145 0.02780.2996 0.4004 0.4588 0.4970 0.150 0.05020.3088 0.4084 0.4663 0.5042 0.155 0.07020.3180 0.4164 0.4738 0.5115 0.160 0.08850.3272 0.4244 0.4813 0.5187 0.165 0.10570.3363 0.4324 0.4888 0.5259 0.170 0.12190.3453 0.4403 0.4963 0.5331 0.175 0.13730.3543 0.4482 0.5037 0.5403 0.180 0.15200.3632 0.4561 0.5112 0.5475 0.185 0.16620.3721 0.4640 0.5186 0.5547 0.190 0.18000.3809 0.4718 0.5260 0.5619 0.195 0.19330.3897 0.4797 0.5334 0.5690 0.200 0.20630.3984 0.4875 0.5408 0.5762 2Q03~34 TABLE B-A
Design Parameters of Electron InterferenCe Filter/Emitter Comprised of Nine Layers in an [H L H L HH L H L H]
Configuration Surrounded by GaO 55Alo 45As and Designed to Emit 0.200 eV Electrons When Biased at 0.100 eV.

Ending Number Unbiased Starting Mono- Mono- Electron Normalize Layer Layer Monolayer layer layers Al Potential Effective 10 Number Type Number Number Thick Comp. Energy Ma*ss j ij-1 ij pj Xj Vj m j/mO

1 H 0 7 7 0.22220.1718 0.0854 2 L 7 16 9 0.41510.3209 0.1015 3 H 16 23 7 0.26630.2059 0.0891 4 L 23 32 9 0.44930.3473 0.1043 HH 32 44 12 0.06390.0494 0.0723 6 L 44 52 8 0.43640.3374 0.1032 7 H 52 58 6 0.14420.1115 0.0790 8 L 58 65 7 0.37480.2898 0.0981 9 H 65 71 6 0.19510.1508 0.0832 TABLE S-I
Upper- and Lower-Cutoff Energies and Range of Energies for the Lowest-Order Waveguide Mode, Mo~ for Various Film SThicknesses in a GaO 85Alo 15As substrate, GaAs film, GaO 70Alo 30As cover ~uantum Well Waveguide Waveguide Film Thickness (GaAs) d (nm) 2.8267 5.6533 8.4800 d (monolayers) 10 20 30 Upper Cutoff EnergY, EUcO 0.6536 0.6926 (eV) Lower Cutoff EnergY, ELco 0.0551 0.0341 (eV) Propagation Energy Range, 0.4006 0.5985 0.6585 E (eV) APPENDIX I
The following illustrates the manner in which the design of electromagnetic optical wave devices is mapped into the design of solid state quantum mechanical electron wave devices.
Using the inventive mapping between quantum mechanical electron waves and electromagnetic optical waves set forth above, in accordance with the present invention, the characteristics of a many-boundary solid state superlattice system can be determined by directly applying the techniques used in electromagnetics to design devices such as interference filters and, in particular, by applying the chain matrix design approach which is well known to those of ordinary skill in the art.
The manner in which this occurs is illustrated as follows. Consider a solid state superlattice structure comprised of M layers which is bounded on one side by an input medium designated as layer 0 and which is bounded on the other side by an output medium designated as layer M+1.
In layer m-l of the superlattice, the amplitude of an electron wave which is traveling in the positive direction and which is incident upon upon layer m is designated as Ui m. In layer m-l, the amplitude of the electron wave which is traveling in the negative direction and which is reflected from layer m is designated as Ur m. These complex wave amplitudes may be expressed in terms of corresponding plitude5 Ui,m+l and Ur,m+l which are incident upon and reflected from the boundary between layers m and m+l as follows:
(AI-l):
Ui,m = _____ 1 re,m exp(jke,mdmCS@m) Ui,m+l Ur,m te,m e,m 1 0 eXp(-jke,mdmcOs@m) Ur,m+l 200~ 4 where te m is the amplitude transmissivity of the interface between layers m-1 and m, re m is the amplitude reflectivity at the same interface, ke m is the magnitude of the electron wavevector in layer m, dm is the thickness of layer m, and @m is the angle of the wavevector direction in layer m. For a superlattice comprised of M layers, the total normalized transmitted electron wave amplitude Ut M+1 in output layer M+l and the total normalized reflected electron wave amplitude Ur 0 in input layer 0 are obtained by chain multiplying a total of M+1 versions of equation AI-l together for each one of the M layers and one for the output layer.
The result is:
(AI-2~:
_M
1 _____ 1 re,m exp(ike mdmCos@m) ur~o te m e,m o eXp(-jke~mdmcos@m) m-_ 1 1 re,M+1 t,m+1 te,m re,M+l 1 0 and this can be solved directly for the electron wave amplitude transmissivity Ut M+l and the electron wave amplitude reflectivity Ur 0.
Equations of the form of equations AI-1 and AI-2 have been widely used over many years for the design of thin film optical coatings and filters. Types of devices that have been treated and for which extensive designs exist include low pass filters, high pass filters, notch filters (narrow band and wide band), impedance transformers (antireflection coatings), and high reflectance surfaces (dielectric mirrors).

Claims (49)

1. A solid state quantum mechanical electron wave device comprising a superlattice structure characterized in that the superlattice structure comprises a multiplicity of adjacent layers of semi-conductor material, each of which layers has a potential energy barrier and supports substantially ballistic electron transport at energies above the potential energy barrier of the layer.

2. The solid state quantum mechanical electron wave device of claim 1 wherein the potential energy barriers, electron effective masses and thicknesses of the layers of the superlattice structure are predetermined so that the device provides a predetermined transmissivity for electrons having kinetic energies in a predetermined range.

3. The device of claim 1 wherein the potential energy barriers, electron effective masses and thicknesses of the layers of the superlattice structure are predetermined so that the device provides a predetermined reflectivity for electrons having kinetic energies in a predetermined range.

4. The device of claim 3 wherein the potential energy barriers, electron effective masses and thicknesses of the layers of the superlattice structure have values which are determined by predetermined analogy to values of optical indices of refraction and thicknesses of layers of an optical wave device comprised of a multiplicity of adjacent layers of dielectric material which provides an optical reflectivity for electromagnetic optical waves which is substantially similar to the predetermined reflectivity for electrons.

5. The device of claim 3 wherein at least one of the layers has a thickness which is substantially equal to an integral multiple of an electron quarter-wavelength in the material comprising the layer for electrons having kinetic energies in the predetermined range.

6. The device of claim 5 wherein the material of the layers alternates between a first semiconductor material and a second semiconductor material.

7. The device of claim 1 wherein at least one layer of the superlattice is comprised of a material selected from binary, ternary and quaternary semiconductor compositions.

8. The device of claim 5 wherein a first end of the superlattice structure is adjacent a layer of semiconductor material having a first predetermined conduction band height and the other end of the superlattice structure is adjacent a layer of semiconductor material having a second predetermined conduction band height.

9. A solid state quantum mechanical electron wave device for guiding electrons which comprises a semiconductor layer disposed between a first and a second superlattice structure, each of which superlattice structures is comprised of a multiplicity of adjacent layers of semi-conductor material, each of which layers has a potential energy barrier and supports substantially ballistic electron transport at energies above the potential energy barrier of the layer and each of which superlattice structures substantially totally reflects electrons having kinetic energy in a predetermined range.

10. A solid state quantum mechanical hole wave device comprising a superlattice structure which is comprised of a multiplicity of adjacent layers of semi-conductor material, each of which layers has a potential energy barrier and supports substantially ballistic hole transport at energies above the potential energy barrier of the layer.

11. The device of claim 2 wherein the potential energy barriers, effective masses and thicknesses of the layers of the superlattice structure have values which are determined by predetermined analogy to values of optical indices of refraction and thicknesses of layers of an optical wave device comprised of a multiplicity of adjacent layers of dielectric material which provides an optical transmissivity for electromagnetic optical waves which is substantially similar to the predetermined transmissivity for electrons.

12. The device of claim 11 wherein at least one of the layers has a thickness which is substantially equal to an integral multiple of an electron quarter-wavelength in the material comprising the layer for electrons having kinetic energies in the predetermined range.

13. The device of claim 12 wherein the material of the layers alternates between a first semiconductor material and a second semiconductor material.

14. The device of claim 12 wherein a first end of the superlattice structure is adjacent a layer of semiconductor material having a first predetermined conduction band height and another end of the superlattice structure is adjacent a layer of semiconductor material having a second predetermined conduction band height.

15. The device of claim 1 wherein at least one of the layers has a thickness which is substantially equal to an integral multiple of an electron quarter-wavelength in the material comprising the layer for electrons having kinetic energies in a predetermined range.

16. The device of claim 14 wherein at least one of the layers has a thickness which is substantially equal to an integral multiple of a hole quarter-wavelength in the material comprising the layer for holes having kinetic energies in a predetermined range.

17. A solid state, quantum mechanical, electron wave filter/emitter which comprises:
a superlattice structure which is comprised of a multiplicity of adjacent layers of semi-conductor material, each of which layers has a potential energy barrier and supports substantially ballistic electron transport at energies above the potential energy barrier of the layer and means for applying a bias potential energy to the superlattice structure wherein the potential energy barriers, electron effective masses and thicknesses of the layers of the superlattice structure are predetermined as if a predetermined bias potential has been applied to the applying means so that the application of the predetermined bias potential to the applying means causes the filter/emitter to function as a filter/emitter for electrons having kinetic energies in a predetermined range.

18. The filter/emitter of claim 17 wherein at least one of the layers of the superlattice structure has a thickness which is substantially equal to an integral multiple of an electron quarter-wavelength in the layer, the electron wavelength being determined as if the predetermined bias potential has been applied to the applying means.

19. The filter/emitter of claim 18 wherein the superlattice structure is comprised of a first reflector, a spacer and a second reflector wherein:
the first reflector is comprised of a predetermined number of adjacent pairs of adjacent layers of the superlattice structure, each layer of the pair having a thickness which is substantially equal to an odd multiple of an electron quarter-wavelength in the layer, the electron wavelength being determined as if the predetermined bias potential has been applied to the applying means;
the spacer is comprised of a layer of the structure having a thickness which is substantially equal to an integral multiple of an electron quarter-wavelength in the layer, the electron wavelength being determined as if the predetermined bias potential has been applied to the applying means; and the second reflector is comprised of a predetermined number of adjacent pairs of adjacent layers of the structure, each layer of the pair having a thickness which is substantially equal to an odd multiple of an electron quarter-wavelength in the layer, the electron wavelength being determined as if the predetermined bias potential has been applied to the applying means.

20. The filter/emitter of claim 19 wherein:
the two layers in each pair of layers of the first reflector have different electron wave amplitude refractive indices and two layers in each layer of the second reflector have different electron wave amplitude refractive indices.

21. The filter/emitter of claim 20 wherein:
a first layer of each pair of layers of the first reflector having a higher electron wave amplitude index of refraction than that of a second layer of the pair, the pairs of layers being arranged so that the first layer is followed by the second layer as one proceeds from a first end of the filter/emitter to a second end of the filter/emitter; and a first layer of each pair of layers of the second reflector having a higher electron wave amplitude index of refraction than that of a second layer of the pair, the pair of layers being arranged so that the second layer is followed by the first layer as one proceeds from the first end to the second end.

22. The filter/emitter of claim 20 wherein:
a first layer of each pair of layers of the first reflector having a higher electron wave amplitude index of refraction than that of a second layer of the pair, the pairs of layers being arranged so that the second layer is followed by the first layer as one proceeds from a first end of the filter/emitter to a second end of the filter/emitter; and a first layer of each pair of layers of the second reflector having a higher electron wave amplitude index of refraction than that of a second layer of the pair, the pair of layers being arranged so that the first layer is followed by the second layer as one proceeds from the first end to the second end.

23. The filter/emitter of claim 18 wherein the superlattice structure is comprised of alternating layers of a first semiconductor material and a second semiconductor material and one end of the filter/emitter is adjacent a layer of semiconductor material having a first predetermined conduction band height and the other end of the filter/emitter is adjacent a layer of semiconductor material having a second predetermined conduction band height.

24. The filter/emitter of claim 23 wherein at least one of the first and second semi-conductor materials is a binary, ternary or quaternary semi-conductor composition of III-V elements.

25. The filter/emitter of claim 17 wherein the layers of the superlattice structure alternately have a first and a second electron wave amplitude index of refraction, the first index of refraction being higher than the second index of refraction, each layer having a thickness which is substantially equal to an integral multiple of a quarter of an electron wavelength in the layer, the electron wave amplitude index of refraction and electron wavelength being determined as if the predetermined bias potential has been applied to the applying means.

26. A solid state quantum mechanical, hole wave filter/
emitter which comprises:
a superlattice structure which is comprised of a multiplicity of adjacent layers of semi-conductor material, each of which layers has a potential energy barrier and supports substantially ballistic hole transport at energies above the potential energy barrier of the layer and means for applying a bias potential energy to the superlattice structure wherein the potential energy barriers hole effective masses and thicknesses of the layers of the superlattice structure are predetermined as if a predetermined bias potential has been applied to the applying means so that the application of the predetermined bias potential to the applying means causes the filter/emitter to function as a filter/emitter for holes having kinetic energies in a predetermined range.

27. The filter/emitter of claim 23 wherein at least one of the first and second semi-conductor materials is a material selected from binary, ternary and quaternary semi-conductor compositions of II-VI elements.

28. The filter/emitter of claim 26 wherein at least one of the layers of the superlattice structure has a thickness which is substantially equal to an integral multiple of an electron quarter-wavelength in the layer, the electron wavelength being determined as if the predetermined bias potential has been applied to the applying means.

29. A method for fabricating an electron wave, semiconductor device comprising a superlattice structure which supports substantially ballistic electron transport at energies above the superlattice potential energy barriers, which method comprises the steps of:
forming an epitaxial superlattice structure comprised of a predetermined number of layers of semiconductor materials;
characterized in that the thickness and the composition of each of the layers are chosen by utilizing an electromagnetic optical wave device design whose parameters are determined in accordance with optical design methods and converting the parameters of the optical device design into the thickness and composition of each of the layers of the superlattice structure by mapping the optical phase index of refraction into a first solid state index of refraction for phase quantities which is proportional to the square root of the product of the electron kinetic energy and the electron effective mass and by mapping the optical amplitude index of refraction into a second solid state index of refraction for amplitude quantities which is proportional to the square root of the electron kinetic energy divided by the electron effective mass.

30. The method of claim 29 which is further characterized in that at least one thickness obtained by converting is varied in odd integer multiples of an electron wave quarter-wavelength until the thickness substantially equals an integer number of monolayer thicknesses of the material of the layer.

31. The method of claim 29 which is further characterized in that at least one composition is varied in order to provide a direct band gap material composition.

32. The method of claim 29 which is further characterized in that the superlattice structure is comprised of a first material A1-xBxC where A, B and C are the chemical elements in the first material and x is the mole fraction of element B in the first material and a second material A1-yByC, wherein:
the conduction band height in the materials is substantially directly proportional to x and y, i.e. V1 = Ax and V2 = Ay, respectively;
the effective electron mass in the materials is substantially linearly related to x and y, i.e. m*1 = (B + Cx)mo and m*2, = (B + Cy)mo, respectively; and x and y are determined from roots of the following equations:
ACx2 + (AB - CEp) x + (h2/32mo) [ (2q1 - 1)2/p1r1 2] - BEp = 0 ACy2 + (AB - CEp) y + (h2/32mo) [ (2q2 - 1)2/p2r2 2] - BEp = 0 wherein Ep is the energy of the electrons which are incident upon the superlattice, h is Planck's constant, mo is the free electron mass, r1 and r2 are the monolayer thicknesses of the materials, d1 and d2 are the thicknesses of the superlattice layers which must be integer multiples of the monolayer thicknesses and d1 = p1r1 and d2 = p2r2 where q1 and q2 are each an integer starting at 1.

33. A method for fabricating a biased, electron wave, semiconductor, filter/emitter comprising a superlattice structure which supports substantially ballistic electron transport at energies above the superlattice potential energy barriers, which method comprises the steps of:
forming an epitaxial superlattice structure comprised of a predetermined number of layers of semiconductor materials;
characterized in that the thickness and the composition of each of the layers are determined in accordance with an iterative method which utilizes as an initial estimate an electromagnetic optical wave device design whose parameters are determined in accordance with optical design methods and converting the parameters of the optical device design into the thickness and composition of each of the layers of the superlattice structure by mapping the optical phase index of refraction into a first solid state index of refraction for phase quantities which is proportional to the square root of the product of the electron kinetic energy and the electron effective mass and by mapping the optical amplitude index of refraction into a second solid state index of refraction for amplitude quantities which is proportional to the square root of the electron kinetic energy divided by the electron effective mass.

34. The method of claim 33 which is further characterized in that the iterative method further comprises determining the thickness of at least one layer of the superlattice structure by determining the wavelength of the layer under bias at each step of the iteration and requiring the thickness of the layer under bias to correspond substantially to an integer number of monolayer thicknesses of the material of the layer.

35. The method of claim 34 which is further characterized in that the iterative method further comprises varying the thickness of at least one layer in odd integer multiples of a quarter-wavelength until the thickness corresponds substantially to an integer number of monolayer thicknesses of the material of the layer.

36. The method of claim 35 which is further characterized in that the iterative method further comprises varying the composition of the material of at least one layer to provide for a direct band gap material composition.

37. The method of claim 36 which is further characterized in that (a) the layers of the superlattice structure comprise materials of the form A1-xBxC where A, B and C are the chemical elements in the superlattice and x is the mole fraction of element B in the superlattice;
(b) the conduction band height in the jth layer is substantially directly proportional to xj, i.e. Vj = Axj;

(c) the effective electron mass in the materials is substantially linearly related to xj, i.e., m*j = (B + Cxj)m0; and (d) the condition that a layer under bias have a thickness which corresponds to a multiple of an electron quarter-wavelength which enters the filter/emitter with a kinetic energy of (KE)in is given by the following equation:
{2L[2mo(B + Cxj)]1/2/3?Vbias}*
{[Vo+(KE)in-Axj+Vbiaszj/L]3/2 - [Vo+(KE)in-Axj+Vbiaszj-1/L]3/2}
= (2qj - 1) ?/2 where L is the thickness of the superlattice structure, zj-1 is the distance to start of the jth layer as measured from the start of the superlattice structure, zj is the distance to the end of the jth layer from the start of the superlattice structure, Vbias is the bias potential energy, qj is an integer, ? is Planck's constant divided by 2?, m0 is the free electron mass, and V0 is the height of the conduction band for the material adjacent the entrance of the superlattice structure.

38. An electron waveguide which comprises a substrate semiconductor layer; a film semiconductor layer adjacent the substrate semiconductor layer; and a cover semiconductor layer adjacent the film semiconductor layer, which device is characterized in that a portion of the substrate semiconductor layer which is adjacent the film semiconductor layer, the film semiconductor layer, and a portion of the cover semiconductor layer adjacent the film semiconductor layer provide substantially ballistic electron transport; and the thickness of the film semiconductor layer and compositions of the semiconductor layers are predetermined to provide a potential well so that electron waveguide modes exist for electron energies in the well or for electron energies above one or both of the potential energy barriers of the substrate semiconductor layer and the cover semiconductor layer, respectively.

39. The waveguide of claim 38 which is further characterized in that the electron potential energy barrier height for the cover semiconductor layer and the substrate semiconductor layer of the semiconductor device are substantially the same.

40. The waveguide of claim 39 which is further characterized in that the cover semiconductor layer and the substrate semiconductor layer of the semiconductor device are comprised of the same material.

41. The waveguide of claim 38 which is further characterized in that the cover semiconductor layer, the film semiconductor layer and the substrate semiconductor layer are fabricated from compositions of the form Ga1-xAlAs where x is the mole fraction of aluminum.

42. The waveguide of claim 41 which is further characterized in that the film semiconductor layer is comprised of GaAs.

43. The waveguide of claim 38 which is further characterized in that the thickness of the film semiconductor layer d is determined so that at least the vth electron waveguide mode propagates, where v is an integer and d is substantially equal to or greater than:
d = (?/2m*f(Vs - Vf)]1/2)*
*(tan-1[m*f(Vc - Vs)/m*c(Vs - Vf)]1/2 + v?) wherein m*c, m*f and m*s are the electron effective masses in the cover semiconductor layer, the film semiconductor layer and the substrate semiconductor layer, respectively; wherein Vc, Vf and Vs are the electron potential energy barrier heights associated with the cover semiconductor layer, the film semiconductor layer and the substrate semiconductor layer, respectively; and ? is Planck's constant divided by 2?.

44. The waveguide of claim 43 wherein the waveguide only supports modes no higher than the vth mode which is further characterized in that K1 * [K2 + v?]< d < K1 * [K2 + (v + 1)?]
where: K1 = ?/[2m*f(Vs - Vf)]1/2 K2 = tan-1[m*f(Vc - Vs)/m*c(Vs - Vf)]1/2

45. The waveguide of claim 39 which is further characterized in that the thickness of the film semiconductor layer d is determined so that at least the vth electron waveguide mode propagates, where v is an integer; and d is substantially equal to or greater than:

d = v??/[2m*f(Vs - Vf)]1/2 wherein m*c, m*f and m*s are the electron effective masses in the cover semiconductor layer, the film semiconductor layer, and the substrate semiconductor layer, respectively; wherein Vc, Vf and Vs are the electron potential energy barrier heights associated with the cover semiconductor layer, the film semiconductor layer and the substrate semiconductor layer, respectively; and ? is Planck's constant divided by 2?.

46. The waveguide of claim 45 wherein the waveguide only supports the lowest order mode, i.e., v = 0, which is further characterized in that 0 < d < ??/[2m*f(Vs - Vf)]1/2

47. The device of claim 38 wherein the semiconductor materials are binary, ternary or quaternary semiconductor compositions of III-V elements or II-VI
elements.

48. An electron waveguide which comprises a substrate semiconductor layer; a film semiconductor layer adjacent the substrate semiconductor layer; and a cover semiconductor layer adjacent the film semiconductor layers, which device is characterized in that the film semiconductor layer provides substantially ballistic electron transport; and the thickness of the film semiconductor layer and compositions of the semiconductor layers are predetermined to provide a potential well so that electron waveguide modes exist for electron energies in the well or for electron energies above one or both of the potential energy barriers of the substrate semiconductor layer and the cover semiconductor layer, respectively.

49. A hole waveguide which comprises a substrate semiconductor layer; a film semiconductor layer adjacent the substrate semiconductor layer; and a cover semiconductor layer adjacent the film semiconductor layer, which device is characterized in that a portion of the substrate semiconductor layer which is adjacent the film semiconductor layer, the film semiconductor layer, and a portion of the cover semiconductor layer adjacent the film semiconductor layer provide substantially ballistic hole transport; and the thickness of the film semiconductor layer and compositions of the semiconductor layers are predetermined to provide a potential well so that hole waveguide modes exist for hole energies in the well or for hole energies above one or both of the potential energy barriers of the substrate semiconductor layer and the cover semiconductor layer, respectively.