Thermal diode for energy conversion

[0002] This application claims priority to U.S. Provisional Patent Application Serial No. 60/213,564, filed on Jun. 22, 2000, and is a continuation in part of U.S. patent application Ser. No. 09/519,640, filed Mar. 6, 2000, which claims priority to provisional application Serial No. 60/123,900, filed Mar. 11, 1999, which applications are incorporated herein by specific reference.

[0003] 1. Field of the Invention

[0004] This invention relates to the conversion of thermal energy to electric energy, and electric energy to refrigeration, and more particularly to a solid state thermionic converter using semiconductor diode implementation.

[0005] 2. Relevant Technology

[0006] Thermionic energy conversion is a method of converting heat energy directly into electric energy by thermionic emission. In this process, electrons are thermionically emitted from the surface of a metal by heating the metal and imparting sufficient energy to a portion of the electrons to overcome retarding forces at the surface of the metal in order to escape. Unlike most other conventional methods of generating electric energy, thermionic conversion does not require either an intermediate form of energy or a working fluid, other than electric charges, in order to change heat into electricity.

[0007] In its most elementary form, a conventional thermionic energy converter consists of one electrode connected to a heat source, a second electrode connected to a heat sink and separated from the first electrode by an intervening space, leads connecting the electrodes to the electric load, and an enclosure. The space in the enclosure is either highly evacuated or filled with a suitable rarefied vapor, such as cesium.

[0008] The essential process in a conventional thermionic converter is as follows. The heat source supplies heat at a sufficiently high temperature to one electrode, the emitter, from which electrons are thermionically evaporated into the evacuated or rarefied-vapor-filled interelectrode space. The electrons move through this space toward the other electrode, the collector, which is kept at a low temperature near that of the heat sink. There the electrons condense and return to the hot electrode via external electric leads and an electric load connected between the emitter and the collector.

[0009] An embodiment of a conventional thermionic converter 100 is schematically illustrated in FIG. 1. These conventional devices typically comprise an emitter 110, or low electron-work-function cathode, a collector 112, or comparatively colder, high electron-work-function anode, an enclosure 114, suitable electric conductors 116, and an external load 118. Emitter 110 is exposed to heat flow 120 which causes this cathode to emit electrons 122, thus closing the electric circuit and providing an electric intensity to load 118. As indicated above, interelectrode space 130 in conventional thermionic converters is an evacuated medium or a rarified-vapor-filled medium.

[0010] The flow of electrons through the electric load is sustained by the temperature difference between the electrodes. Thus, electric work is delivered to the load.

[0011] Thermionic energy conversion is based on the concept that a low electron work function cathode in contact with a heat source will emit electrons. These electrons are absorbed by a cold, high work function anode, and they can flow back to the cathode through an external load where they perform useful work. Practical thermionic generators are limited by the work function of available metals or other materials that are used for the cathodes. Another important limitation is the space charge effect. The presence of charged electrons in the space between the cathode and anode will create an extra potential barrier which reduces the thermionic current. These limitations detrimentally affect the maximum current density, and thus present a major problem in developing large-scale thermionic converters.

[0012] Conventional thermionic converters are typically classified as vacuum converters or gas-filled converters. Vacuum converters have an evacuated medium between the electrodes. These converters have limited practical applications.

[0013] Embodiments in a first class of gas-filled converters are provided with a vaporized substance in the interelectrode space that generates positive ions. This vaporized substance is commonly a vaporized alkali metal such as cesium, potassium and rubidium. Because of the presence of these positive ions, liberated electrons can more easily travel from the emitter to the collector. The emitter temperature in these types of conventional devices is in part determined by the vaporization temperature of the substance that generates the positive ions. Generally, the emitter temperature should be at least 3.5 times the temperature of the reservoir of the positive ion generating substance if efficient production of ions is to be achieved in these conventional devices.

[0014] Embodiments in a second class of gas-filled converters are provided with a third electrode to generate ions. The gas in the interelectrode space in these conventional devices is an inert gas such as neon, argon and xenon. Although these converters can operate at lower temperatures, such as about 1500 K, they are more complex.

[0015] Typical conventional thermionic emitters are operated at temperatures ranging from 1400 to 2200 K and collectors at temperatures ranging from 500 to 1200 K. Under optimum conditions of operation, overall efficiencies of energy conversion range from 5 to 40%, electric power densities are of the order of 1 to 100 watts/cm², and current densities are of the order of 5 to 100 A/cm². In general, the higher the emitter temperature, the higher the efficiency and the power and current densities with designs accounting for radiation losses. The voltage at which the power is delivered from one unit of a typical converter is 0.3 to 1.2 volts, i.e., about the same as that of an ordinary electrolytic cell. Thermionic systems with a high power rating frequently consist of many thermionic converter units connected electrically in series. Each thermionic converter unit is typically rated at 10 to 500 watts.

[0016] The high-temperature attributes of thermionic converters are advantageous for certain applications, but they are restrictive for others. This is because the required emitter temperatures are generally beyond the practical capability of many conventional heat sources. In contrast, typical thermoelectric converters are operable at heat source temperatures ranging from 500 to 1500 K. However, even under optimum conditions, overall efficiencies of thermoelectric energy converters only range from 3 to 10%, electric power densities are normally less than a few watts/cm², and current densities are of the order of 1 to 100 A/cm².

[0017] From a physics standpoint, thermoelectric devices are similar to thermionic devices. In both cases a temperature gradient is placed upon a metal or semiconductor, and both cases are based upon the concept that electron motion is electricity. However, the electron motion also carries energy. A forced current transports energy for both thermionic and thermoelectric devices. The main difference between thermoelectric and thermionic devices is in the transport mechanism: ballistic and diffusive transport for thermionics and ohmic transport for thermoelectrics. Ohmic flow is microscopically diffusive, but not macroscopically so. The distinguishing feature is whether excess carriers are present. In thermoelectrics, the carriers normally present are responsible for current. In thermionics, the current is due to putting excess carriers in the gap. A thermionic device has a relatively high efficiency if the electrons ballistically go over and across the gap. For a thermionic device all of the kinetic energy is carried from one electrode to the other. The motion of electrons in a thermoelectric device is quasi-equilibrium and ohmic, and can be described in terms of a Seebeck coefficient, which is an equilibrium parameter.

[0018] In structures with narrow barriers, the electrons will not travel far enough to suffer collisions as they cross the barrier. Under these circumstances, the ballistic version of thermionic emission theory is a more accurate representation of the current transport. The current density is given by: $j=A_0{T}^2{e}^{\frac{-e\varphi }{k_{B}T}},$

[0019] where A₀is the Richardson's constant, φ is the barrier height (electron work function), e is the electron charge, k_Bis Boltzmann's constant, and T is the temperature. Richardson's constant A₀is given by A₀=(emk_B²T²)/(2π²h²), where m is the effective electron mass and h is reduced Plank's constant.

[0020] The foregoing electron current density equation provides quantitative language for explaining some of the observations described above. For example, this equation for the emission current shows that the rate of emission increases rapidly with temperature, and it decreases exponentially with the work function.

[0021] Solutions to the foregoing problems have been sought according to the present state of the art by using vacuum converters or gas-filled converters. Attempts to reduce space-charge effects with vacuum converters have involved the reduction of the interelectrode separation to the order of micrometers. Attempts to reduce the same effects with gas-filled converters have led to the introduction of positive ions into the cloud of electrons in front of the emitter. Nevertheless, these conventional devices still present shortcomings such as those related to limited maximum current densities and temperature regimes. Consequently, there remains a need to provide a more satisfactory solution to converting thermal energy to electric energy at lower temperature regimes with high efficiencies and high power densities.

[0022] The present invention was developed to fill a need for a device which efficiently converts thermal energy to electric energy at relatively low operating temperatures and with power densities and efficiencies high enough for commercial applications. The present invention also operates in reverse mode to provide efficient cooling.

[0023] The present invention seeks to resolve a number of the problems which have been experienced in the background art, as identified above. More specifically, the apparatus and method of this invention constitute an important advance in the art of thermionic power conversion, as evidenced by the following objects and advantages realized by the invention over the background art.

[0024] For example, one object of the present invention is to generate high power densities and efficiencies of a typical thermionic converter, but to operate at temperature regimes of typical thermoelectric devices.

[0025] A further object of the present invention is to perform energy conversion through thermionic emission using semiconductor thermal diode implementation.

[0026] Additional objects and advantages of the invention will be apparent from the description which follows, or may be learned by the practice of the invention.

[0027] Briefly summarized, the foregoing and other objects are achieved by a solid state thermionic converter utilizing semiconductor diode implementation comprising an emitter that preferably comprises an n*-type region; a gap region between the emitter and a collector, the gap region being preferably adjacent to said n*-type region; and a cold ohmic contact connected to said gap region, said cold ohmic contact having a recombination collector region formed between said cold ohmic contact and said gap region. In some embodiments of this invention a collector provides a recombination element and such collector is in electric communication with a cold ohmic contact. Preferably, a hot ohmic contact is in electric communication with the emitter. The gap region maybe n-type, p-type, or intrinsic. To use the electric current generated by embodiments of the converter of this invention, the electric circuit is typically closed externally with an electric load, preferably connected to the hot ohmic contact and the cold ohmic contact. It is understood that terms such as “electric communication”, “electric connection” and “electric contact” refer to a relationship between elements whereby electric current can flow between such elements, whether such elements are in direct contact or the electric current flow is facilitated by at least one conductor linking such elements.

[0028] In the refrigeration embodiment, carrier transport is assisted by an external electric field. A first ohmic contact on the emitter, preferably comprising an n*-type region, is connected to a thermal load that is cooled by heat flow from electrons leaving the emitter. As described in connection with preferred embodiments of heat-to-electricity converters according to this invention, electrons in refrigeration embodiments preferably circulate from the emitter, preferably from a hot ohmic contact on the preferably n*-type region, to the gap region. A gap region is preferably adjacent to the emitter, and a second ohmic contact having a recombination collector region is preferably formed between the second ohmic contact and the gap region. The gap region in embodiments of this invention may be n-type, p-type, or intrinsic. A heat exchanger dissipates the heat from hot electrons on the second ohmic contact connected to the gap region.

[0029] In order to more fully understand the manner in which the above-recited advantages and objects of the invention are obtained, a more particular description of the invention will be rendered by reference to specific embodiments thereof which are illustrated in the appended drawings. Understanding that these drawings depict only typical embodiments of the invention and are therefore not to be considered limiting of its scope, the presently preferred embodiments and the presently understood best mode of the invention will be described with additional detail through use of the accompanying drawings in which:

[0030] FIG. 1 schematically shows an embodiment of a conventional thermionic converter.

[0031] FIG. 2 is a cross-sectional view of a thermal diode of the present invention.

[0032] FIG, 3 is a plot of the efficiency as a function of the normalized conductivity χ for the case of Δτ=0.3.

[0033] FIG. 4 is a plot of the efficiency as a function of the normalized conductivity χ for the case of Δτ=0.5

[0034] FIG. 5 is a plot of the normalized conductivity parameter χ as a function of temperature for InSb, assuming that N_D*D=10²⁰cm⁻³.

[0035] FIG. 6 is a plot of the intrinsic carrier density for InSb between 300 K and 600 K.

[0036] FIG. 7 is a plot of the normalized potential vs x for model doping concentrations with a linear temperature gradient.

[0037] FIG. 8 illustrates the induced EMF between the emitter and collector for an idealized InSb symmetric structure with N_d⁺=10²⁰.

[0038] FIG. 9 illustrates the computed thermopowers for electrons and holes for a thermal diode example with N_a⁻=5×10¹⁸.

[0039] FIG. 10 shows electron and hole concentrations for an n*pn* thermionic structure in InSb, where the donor concentration in the emitter and collector regions is 10²⁰cm⁻³, and the acceptor concentration in the gap region is 10¹⁷cm⁻³.

[0040] FIG. 11 shows electron concentrations for an n*nn* thermionic structure in InSb, where the donor concentration in the emitter and collector regions is 10²⁰cmn⁻³, and the donor concentration in the gap region is 10¹⁴cm⁻³.

[0041] FIG. 12 shows electron and hole concentrations for an n*nn* thermionic structure in InSb, where the donor concentration in the emitter and collector regions is 10²⁰cm⁻³, and the donor concentration in the gap region is 8×10¹⁷cm⁻³.

[0042] FIG. 13 shows the normalized barrier height Δu=qΔφ/k_BT_maxas a function of the doping concentration.

[0043] FIG. 14 shows short circuit current as a function of normalized barrier height for the structures considered above with T_max=600 K and Δτ=0.5.

[0044] FIG. 15 shows current and voltage characteristic for a 625 μ thick InSb design, with an emitter electron concentration of 10²⁰electrons/cm³, an emitter temperature T_max=600 K and a collector temperature T_min=300 K.

[0045] FIG. 16 shows the load power per unit area for a 625 μ thick InSb design, with an emitter electron concentration 10²⁰electrons/cm³, an emitter temperature T_max=600 K and a collector temperature T_min=300 K.

[0046] FIG. 17 shows the thermal power dissipated per unit area as a function of voltage for a 625 μ thick InSb design, with an emitter electron concentration 10²⁰electrons/cm³, an emitter temperature T_max=600 K and a collector temperature T_min=300 K.

[0047] FIG. 18 shows the efficiency as a function of voltage for a InSb design. Calculations are shown for gap donor densities of 10¹⁷(the lowest curve on the plot), 3×10¹⁷, 5×10¹⁷and 7×10¹⁷(the highest curve on the plot) in units of cm⁻³.

[0048] FIG. 19 shows the results of a numerical optimization of efficiency as a function of gap doping over a wide range of doping densities at a fixed emitter ionized dopant concentration of 10²⁰cm⁻³.

[0049] FIG. 20 shows the results for the optimum efficiency as a function of emitter doping with fixed gap doping N_D=7×10¹⁷cm⁻³.

[0050] FIG. 21 shows the thermal and load power per unit area for thermal diode designs.

[0051] FIG. 22 shows the efficiency of a design with an emitter electron concentration of 10²⁰electrons/cm³and a gap donor density of 7×10¹⁷cm⁻³.

[0052] FIG. 23 shows the optimized efficiency as a fraction of the thermodynamic limit.

[0053] FIG. 24 shows the thermal power flow under conditions of optimum energy conversion at different temperatures.

[0054] FIG. 25 is a cross-sectional view of a compensated thermal diode.

[0055] FIG. 26 shows the current as a function of gap doping for an InSb thermal diode design.

[0056] FIG. 27 shows optimization of efficiency as a function of gap doping with p-type compensation using N_a⁻ concentrations of 7×10¹⁷, 10¹⁸, 2×10¹⁸and 3×10¹⁸cm⁻³.

[0057] FIG. 28 shows a cross-sectional view of a single compensated thermal diode with increasing temperature indicated by the arrow labeled T.

[0058] FIG. 29 illustrates an embodiment having four stacked diodes.

[0059] FIG. 30 illustrates an embodiment having multiple stacked diodes having a curved boundary and forming a wedge-shaped geometry.

[0060] FIG. 31 illustrates an embodiment of stacked diodes wherein the stack boundary approximates an ideal curve as shown in FIG. 30.

[0061] FIG. 32 shows the efficiency under optimized conditions as a function of emitter temperature for different gap doping with perfect compensation. An InSb compensated thermal diode structure 625 μ thick is assumed, with an emitter electron density of 10²⁰cm⁻³and a collector temperature of 300 K.

[0062] FIG. 33 shows the efficiency normalized to the thermodynamic limit under optimized conditions as a function of emitter temperature for the different cases shown in FIG. 32.

[0063] FIGS. 34 and 35 show the dose needed to create a compensated layer over a wide range of ion energies in an n-type InSb diode doped to a concentration n.

[0064] FIG. 36 shows the results for an ohmic contact implantation dose required to achieve a 10²¹cm⁻³shallow doping of Te for an InSb design.

[0065] FIG. 37 shows the ion range for FIG. 36.

[0066] FIG. 38 shows the results for Ag doping to achieve an ohmic contact.

[0067] FIG. 39 shows the results for Ag doping to achieve an ohmic contact.

[0068] FIG. 40 shows the temperature behavior of an InSb gap.

[0069] FIG. 41 shows the temperature dependence on barrier height for an interface layer doped with Te to 3×10¹⁹cm⁻³deposited on InSb doped with Te to 1×10¹⁸cm⁻³with an In emitter.

[0070] FIG. 42 A shows the surface states of a metal-semiconductor contact.

[0071] FIG. 42B schematically illustrates an embodiment of the present invention comprising a metal-semiconductor-interface-barrier reduction layer.

[0072] FIG. 43 shows the I-V curves for a single diode and a stack of three InSb diodes.

[0073] FIG. 44 illustrates a thermal diode for providing cooling.

[0074] FIG. 45 shows a compensated thermal diode for providing cooling.

[0075] FIG. 46 shows a graph of one plate efficiency for InSb as a function of Ar ion implantation dose for a sample whose size is 0.50×1.0×1.5 mm³.

[0076] FIG. 47 shows a graph of the ⁴He ion implantation range as a function of ion energy for an InSb target.

[0077] FIG. 48 shows the results of a simulation for the number of vacancies per ion as a function of ⁴He ion energy for the ion implantation referred to in FIG. 47.

[0078] FIG. 49 shows a graph of output current density for an embodiment comprising a Hg_0.86Cd_0.14Te sample as a function of the hot side temperature for the sample with a Cu emitter layer and for the sample with an In-Ga emitter layer.

[0079] FIG. 50 shows a graph of output current density for an embodiment comprising a Hg_0.86Cd_0.14Te sample as a function of the hot side temperature for the sample with an Al substrate and for the sample with an In-Ga substrate.

[0080] FIG. 51 shows a graph of the absolute efficiency as a function of temperature for an embodiment of a thermal diode without compensation comprising a Hg_0.86Cd_0.14Te sample.

[0081] FIG. 52 shows a graph of the efficiency, expressed as a percentage of ideal Carnot cycle efficiency, as a function of temperature for the same embodiment referred to in FIG. 51.

[0082] FIG 53 shows a graph of the absolute efficiency for an embodiment of a sandwich converter as a function of hot plate temperature.

[0083] FIG. 54 shows a graph of the efficiency, expressed as a percentage of ideal Carnot cycle efficiency, as a function of hot plate temperature for the same embodiment referred to in FIG. 53.

[0084] FIG. 55 shows a graph for the Hg_1−xCd_xTe normalized figure of merit relative to that of InSb as a function of x.

[0085] The present invention embodies a solid state thermionic energy converter 10, generally illustrated in FIG. 2, and is directed to a method and apparatus for the conversion of energy. A preferred embodiment of the inventive solid state thermionic energy converter 10 comprises a semiconductor diode having an n*-type region 14 as an emitter, a gap region 16 adjacent to the n*-type region 14, a hot ohmic contact 12 connected to said n*-type region 14, and a cold ohmic contact 20 being a collector and connected to said gap region 16. The cold ohmic contact 20 has a recombination collector region 18 formed between said cold ohmic contact 20 and said gap region 16.

[0086] The recombination region in some embodiments of this invention comprises a distinct layer. In other embodiments of this invention the recombination region is obtained by treating and/or damaging the surface of an ohmic contact or collector. Forming a recombination region in the context of this invention thus includes procedures for incorporating a recombination layer and procedures for treating and/or damaging the surface of an ohmic contact or collector.

[0087] The terms n*-region are used herein to refer to an n-region which has a higher electron concentration than a n-region. Illustrative embodiments of materials comprised in n*-regions are given below. A general characterization of the n*-region and n-region in terms of their relative donor number densities is provided hereinbelow (see Equation (2.22) and its discussion). Examples of n-type regions are provided by regions that include InSb doped with Te at a concentration from about 10¹⁶cm⁻³to about 10¹⁹cm⁻³. Concentrations in the order of 10²⁰cm⁻³are also envisaged as characterizing the dopant concentration of materials in n-type regions in embodiments of this invention. Examples of n*-type regions are provided by regions that include InSb doped with Te at a concentration from about 10¹⁹cm⁻³to about 3·10¹⁹cm⁻³. Concentrations of about 3·10²⁰cm⁻³are also envisaged as characterizing the dopant concentration of materials in n*-type regions in embodiments of this invention. In addition to Te, dopants in some other embodiments of the present invention include at least one of S, Se, and Sn. Furthermore, the symbol n** is used herein to refer to an n-region which has a higher electron concentration than an n*-region. Examples of n**-type regions are provided by regions that include a material such as In, Te, Ga, and Fe.

[0088] An electric load R_Lconnected to hot ohmic contact 12 and to cold ohmic contact 20 is provided with the electric intensity generated by an embodiment of a converter according to this invention. The emitter may be a metal. The gap region 16 may be either moderately doped n-type, p-type, or intrinsic. Electrons are collected in the recombination collector region 18. The heated emitter relative to the collector generates an EMF which drives current through a series load.

[0089] It is also important to note that the inventive principle works for hole conductivity, as well as for electrons. Also, reference to metals herein includes alloys.

[0090] In contrast with conventional thermionic devices, embodiments of converters according to the present invention are solid-state devices. The prior art, however, teaches devices that rely on an evacuated interelectrode space or on a gas-filled interelectrode space. General characteristics of these conventional devices have been summarized above.

[0091] Instead of an evacuated or gas-filled space, embodiments of the present invention incorporate a semiconducting material. Semiconductors are valuable, not for their conductivity, but for two unusual properties. First, the concentration of free carriers, and consequently the conductivity, increases exponentially with temperature (approximately 5% per degree Celsius at ordinary temperatures). Second, the conductivity of a semiconductor can be increased greatly, and to a precisely controlled extent, by adding small amounts of impurities in the process called doping. Since there are two types of mobile charge carriers (electrons and holes), of opposite sign, extraordinary distributions of charge carriers can be created. The semiconductor diode utilizes this property. Semiconductors, pure or doped, p-type, or n-type, are bilateral; current flows in either direction with equal facility. If, however, a p-type region exists in close proximity to an n-type region, there is a carrier density gradient that is unilateral; current flows easily in one direction only. The resulting device, a semiconductor diode, exhibits a very useful control property of carrier transport that can be utilized for energy conversion.

[0092] The following written description and graphic material refer to analytic and numerical models of phenomena that are associated with working embodiments of the present invention. These models are not provided as limiting explanations of the present invention. It is understood that the present invention is not limited or restricted to any single explanation of its underlying physical processes. The purpose of setting forth the analytic and numerical models discussed below is to highlight relevant variables that can be used to design additional embodiments envisaged within the scope of the present invention, even though such embodiments are not explicitly referred to in the context of this written description. With these design tools, the teachings of this written description, and ordinary skill in the art, additional embodiments that are within the scope of the present invention and claims can be designed. Accordingly, the following written description and graphic material describe embodiments of the present invention and provide models that can be used for designing additional embodiments envisaged within the scope of the present invention.

[0093] The models set forth below include analytic and numerical models, and the implementation of such models to specific materials and embodiments of the present invention is also described below. It is understood that the headings in the following material are provided as guides for organization purposes and not as limiting or restrictive statements regarding this written description and figures, which are to be interpreted in their entirety as a whole.

[0094] The analytic model discussed below includes the derivation of expressions for the efficiency and the normalized current-voltage relation. It is also shown in the context of this model that the efficiency is a function of the dimensionless conductivity, which is in turn a function of material parameters and peak emitter doping. Material parameters are provided for a plurality of materials, and it is shown that InSb is a preferred material and that other materials can be used as semiconductors in embodiments of the present invention.

[0095] The numerical model discussed below provides a framework in which carrier injection from the heated emitter region into the gap region is studied. In addition to terminology and numerical solution techniques, basic equations in this model are introduced and solved.

[0096] Results for an InSb thermal diode are presented and discussed following the introduction of the analytical and numerical models because InSb is one of the preferred semiconductor materials for embodiments of the present invention. The behavior of the InSb embodiment is shown to be consistent with the injection of carriers from the emitter into a gap region, with allowed transport across the gap to the collector. These results are consistent with the efficiency depending on gap doping, as the gap doping determines barrier height and current flow. These results also show that efficiency optimization of a thermal diode according to the present invention using InSb can reach 5.5% with a 600 K emitter and an emitter electron density of 10²⁰cm⁻³.

[0097] The following written description and figures also disclose compensation as a technique to increase efficiency in embodiments of the present invention. Compensation includes return current suppression. Methods for forming ohmic contacts in embodiments of the present invention are subsequently described.

[0098] Examples of embodiments of the present invention that comprise InSb with a compensation layer include InSb wafers with a n-type dopant, such as Te, and an emitter layer of Te implanted by a technique such as magnetron sputtering. The compensation layer in these embodiments is formed by p-type impurity implantation. This p-type impurity comprises at least one type of ions such as Ar and He ions, which compensate for the n-type dopant.

[0099] A more preferred material to build an n*/n emitter according to this invention comprises Hg_1−xCd_xTe. For example, a Hg_0.86Cd_0.14Te wafer is used in embodiments of this invention to build a n*/n emitter by reacting Hg_0.86Cd_0.14Te with a n-type impurity substrate such as Al and In-Ga thus creating an electron injecting n* region. A preferred form of an In-Ga material for this purpose is In_0.75Ga_0.25. Embodiments with this emitter exhibit an output electric current density that increases as a function of the hot side temperature. It is shown below that these embodiments attain efficiencies that are above 30% of an ideal Carnot cycle efficiency.

[0100] Hg_1−xCd_xTe is part of a multi-plate or sandwich configuration in other embodiments of this invention. For example, an embodiment of these sandwiches comprises an InSb plate doped with an n-type material such as Te and an emitter layer of InSb sputtered with Te and coated with a material such as In-Ga, preferably In_0.75Ga_0.25.A second plate in this sandwich material comprises Hg_1−xCd_xTe, where x is preferably 0.14.

[0101] Examples of embodiments according to the present invention include design parameters, uncompensated thermal diodes, compensated thermal diodes and Schottky diodes. Furthermore, converters according to the present invention include converters to convert thermal energy to electricity and refrigeration embodiments. As discussed hereinbelow, two types of embodiments include the same main components, whether they operate as thermionic converters for refrigeration or as thermal diodes for converting thermal energy into electricity.

[0102] It has been found in the context of this invention that Hg_1−xCd_xTe with x being from about 0.08 to about 0.15 exhibits a high thermionic figure of merit while remaining semiconductor and allowing an n* emitter layer/compensation layer design and behavior as described herein with respect to other materials. Furthermore, it has also been discovered in the context of this invention that Hg_0.92Cd_0.18Te behaves as an excellent thermoelectric material.

[0103] 1. The Solid State Thermionic Converter

[0104] The basic concept for the present invention is that a highly doped n* region 14 can serve as an emitter region, from which carriers can be boiled off into a gap region 16. The n* region is a semiconductor doped with a high concentration of donor (providing electrons) impurity. For example, InSb can be doped with Te or S. By itself, the dopant does not guarantee n-type conductivity, because it is positioned on quantum levels below the conductance band. For Te doping, these levels are spaced at an energy E_i˜50 meV from the bottom of the conductance band. As set forth in Sze, S. M., Physics of Semiconductor Devices.N.Y., John Wiley & Sons, 1981, pp. 16-27, work that is incorporated herein by reference, the ionized dopant concentration can be estimated for the case when the number of donor impurities N_Dis much larger than the number of acceptor impurities as follows: $n\cong \frac{1}{\sqrt{2}}{\left(N_D{N}_C\right)}^{1/2}{e}^{\frac{-E_D}{2k_{B}T}}$

[0105] where T is the temperature, k_Bis Boltzmann's constant, E_Dis the impurity level spacing from the bottom of the conduction band or donor ionization energy, N_Dis the donor ionization energy, and N_Cis the effective density of states in the conduction band. The effective density of states is given by $N_C=2{\left(\frac{2\pi m^{*}{k}_{B}T}{h^2}\right)}^{3/2}{M}_c,$

[0106] where h is Plank's constant, m* is the electron effective mass, and M_cis the number of equivalent minima in the conduction band.

[0107] InSb band structure, for example, has two such minima. The spacing between the absolute minimum and the next one is about 0.5 eV, and at low temperatures M_cis effectively unity. At temperatures higher than about 600 K, the second minimum becomes significant and M_cis not unity.

[0108] These ionized electrons can be thermally injected into the gap region 18 through the interface barrier Φ_Band create a current i on an external load R_L. The i·R creates a voltage V which works against electron transport as an exponential factor $e^{\frac{-\mathrm{qV}}{\mathrm{kT}}}.$

[0109] The barrier height Φ_Balso gives an exponential factor $e^{\frac{-q{\Phi }_B}{k_{B}T}}.$

[0110] Assuming a purely diffusive current, the pre-exponential factor in the forward current will be $A\frac{{\mathrm{qN}}_i^{*}{D}_N}{L},$

[0111] where D_Nis the diffusion coefficient, A is the area factor, and L is the diffusion length. The resulting forward current is $\begin{array}{cc}{i}_F\sim \frac{{\mathrm{AqN}}_i^{*}{D}_N}{L}{e}^{\frac{-q{\Phi }_B}{k_B{T}_1}}{e}^{\frac{-\mathrm{qV}}{k_B{T}_1}}.& \left(1.1\right)\end{array}$

[0112] This formula presumes a short gap region with linear diffusion. As the gap may be several hundred microns thick, there is the possibility of recombination effects occurring. This effect must be minimized. While recombination will result in a sharper electron profile and increase the prefactor, the price to be paid is high since the energetics associated with the replacement of the electron current with a hole current is unfavorable. The emitter work function is Φ. The reverse current from the collector to the emitter is $\begin{array}{cc}{i}_R=-\frac{{\mathrm{AqD}}_n{N}_D^{*}}{L}{e}^{\frac{-q\Phi }{k_B{T}_0}}.& \left(1.2\right)\end{array}$

[0113] T_ois the collector temperature. It is assumed that the interior area of the emitter is the same as the total area of the collector. It is also assumed that the collector and emitter regions are the same, so that the various prefactors are the same.

[0114] The total current i is composed of the forward and reverse currents, i=i_F+i_R. Making the approximation that the temperature dependence of the prefactors can be neglected, the diode law is $\begin{array}{cc}i=\frac{{\mathrm{AqD}}_n{N}_D^{*}}{L}\left[e^{\frac{-q\left(\Phi +V\right)}{k_B{T}_1}}-e^{\frac{-q\Phi }{k_B{T}_o}}\right].& \left(1.3\right)\end{array}$

[0115] This equation can be viewed as an iV characteristic which is a function of two temperatures. When coupled with the load, there is in addition V=iR_L. These two separate equations define the operating point current and voltage, both of which are functions of the two temperatures.

[0116] The heat flow due to carrier diffusion is on the order of $\begin{array}{cc}{P}_T=i_F\left(\Phi +V+\frac{3k_B{T}_1}{2q}\right)-i_R\left(\Phi +V+\frac{3k_B{T}_o}{2q}\right).& \left(1.5\right)\end{array}$

[0117] This carrier contribution to the heat flow can be thought of intuitively as follows: heat flow is generated one carrier at a time, and the focus is on the heat flow out of the emitter region to the collector. An electron that travels from the emitter to the gap must overcome the emitter-gap barrier Φ as well as the load voltage V, which will result in a local cooling of the emitter. This implies thermal power flow of i_F(Φ+V) out of the emitter. An electron that is viewed as part of the reverse current heats the emitter, and the associated local power into the emitter is −i_R(Φ+V). Electrons thermalize as they diffuse, so they are assumed always to be locally in equilibrium with the temperature of the semiconductor. Consequently, net current flow from one side to the other must also involve a net transfer of thermal energy that is proportional to the current flow and temperature difference. The mean carrier kinetic energy is $\frac{3}{2}{k}_{B}T,$

[0118] so that the net associated thermal power flow is $\frac{3}{2}{k}_{B}T\left(i_F-i_R\right).$

[0119] The phononic thermal power flow due to Fick's conduction law is defined as P_k=K(T1−T₂). This assumes a small temperature difference, as well as a constant thermal conductivity. The efficiency of this system is then $\eta =\frac{P_{\kappa }}{P_T+P_{\kappa }},$

[0120] where as indicated above, P_Tis the heat flow due to carrier diffusion, and P_kis the phononic thermal power flow as given by Fick's conduction law.

[0121] A. Parameterization of the basic model

[0122] In order to identify the fundamental parameters of the system, the first task is to solve the iV characteristic. The current is eliminated to obtain $\begin{array}{cc}V=\frac{R_L{\mathrm{AqD}}_n{N}_D^{*}}{L}\left[e^{\frac{-q\left(\Phi +V\right)}{k_B{T}_1}}-e^{\frac{-q\Phi }{k_B{T}_o}}\right].& \left(1.6\right)\end{array}$

[0123] This provides for identification of the important system parameters. The reduced potential and reduced voltage are defined according to $\phi =\frac{q\Phi }{k_B{T}_o}\mathrm{and}v=\frac{\mathrm{qV}}{k_B{T}_1}.$

[0124] The dimensionless resistance ξ is identified according to $\xi =\frac{R_L{\mathrm{Aq}}^2{D}_n{N}_D^{*}}{{\mathrm{Lk}}_B{T}_1}.$

[0125] In terms of these parameters, the relation is $v=\xi \left[e^{\frac{-\phi T_o}{T_1}}{e}^{-v}-e^{-\phi }\right],$

[0126] which can be recast as $v={\mathrm{\xi e}}^{-\phi }\left[e^{\frac{\phi \left(T_1-T_o\right)}{T_1}}{e}^{-v}-1\right].$

[0127] The absolute temperature difference Δτ is defined according to $\Delta \tau =\frac{T_1-T_o}{T_1}.$

[0128] This leads then to the constraint v=ξe^−φ[e^φΔre^−v−1].

[0129] This normalized current-voltage relation is the same as that obtained in the idealized free streaming limit. Apparently, it is not a critical feature of the scheme that the electrons be free-streaming, as the diffusive model has an identical normalized current and voltage relation.

[0130] B. Efficiency including conduction loss effects

[0131] In the absence of conduction loss, this model corresponds closely to the free streaming limit of a thermionic model. All of the efficiency results apply to this model as well, as long as the conduction loss is not included. Consequently, attention is turned to a consideration of the degradation of the efficiency by conduction loss.

[0132] As shown above, the efficiency including conduction loss according to the simple model proposed here is $\eta =\frac{P_{\kappa }}{P_T+P_{\kappa }}.$

[0133] In terms of dimensionless quantities, this can be rewritten as $\eta =\frac{v}{\begin{array}{c}\phi \left(1-\Delta \tau \right)+v+\left(3/4\right)\left(2-\Delta \tau \right)+\\ \left(3/4\right)\Delta \mathrm{\tau coth}\left[\left(\phi \Delta \tau -v\right)/2\right]+\chi \Delta \frac{\tau }{e^{-\phi }\left[e^{\phi \Delta \tau }{e}^{-v}-1\right]}\end{array}}.$

[0134] In this equation, the dimensionless conductivity is $\chi =\frac{\kappa }{k_B{N}_D^{*}{D}_n}.$

[0135] The efficiency as a function of the normalized conductivity χ is illustrated in FIG. 3 for the case of Δτ=0.3, for different values of the normalized barrier φ. One sees that when the barrier is low, the system is more robust against thermal and electric losses, but the peak efficiency is correspondingly low. If the barrier is high, then the system is not robust against thermal loss, and a high efficiency becomes possible only if the thermal conduction is somehow very small. Consequently, one faces a compromise in which a moderate barrier will likely be optimum in the presence of relatively low loss. FIG. 4 illustrates the results for Δτ=0.5.

[0136] C. Material Parameters

[0137] The results of the discussion above indicates that the maximum efficiency that can be obtained is a function of the dimensionless conductivity χ, which is in turn a function of the material parameters and peak emitter doping. Before proceeding, the relevant material parameters need to be examined to determine whether the available semiconductors allow operation in physically interesting regimes.

[0138] Table 2 lists material parameters and the associated estimates of χ. Values for the normalized conductivity χ are estimated using N*_D=10²⁰cm⁻³. One observes a wide range (almost four orders of magnitude) of possible values of the normalized conductivity χ. Data for the construction of this table were gathered from the CRC Handbook of Chemistry and Physics,₆₇th edition. Materials with small χ are more preferred. According to this criterion, a preferred material among the semiconductors listed in Table 2 is seen to be mercury selenide, with an associated value of χ=0.014. According to FIG. 4, an efficiency that is optimized with respect to the barrier height will be able to reach about 13.3% for φ=5 in the case of Δτ=0.3. From FIG. 4, the corresponding maximum is 23.8% for φ=5 in the case of Δτ=0.5. These numbers are close to half of the thermodynamic maximum values.

[0139] Both the mobility and the thermal conductivity are, in general, functions of temperature. In InSb, both the mobility and the conductivity decrease with increasing temperature. The resulting temperature dependence of χ is illustrated in FIG. 5. FIG. 5 shows the normalized conductivity parameter χ (also referred to as “figure of merit”) as a function of temperature for InSb, assuming that N*_D=10²⁰cm⁻³. Results are shown for n-type gap regions with doping densities N_D=10¹⁵, 10¹⁶, 10¹⁷, 10¹⁸in units of cm⁻³. The normalized conductivity is seen to decrease at higher temperature. In addition, the normalized conductivity decreases in the presence of background carriers, since the mobility of electrons in InSb decreases with increasing doping density.

[0140] Semiconductors listed in Table 2 are examples that include materials that can be used in embodiments of the present invention. InSb is a preferred material among these semiconductors. As it will be shown hereinbelow, Hg_1−xCd_xTe is a more preferred semiconductor which exhibits a normalized conductivity parameter of about half the value of the same parameter for HgSe.

[0141] 2. A Numerical Model in One Dimension

[0142] As stated previously, the emitter region is heated and carriers are injected into the gap region. The associated scale lengths are sufficiently large that the carrier flow in the gap region will be either by drift or by diffusion.

[0143] As indicated in the previous subsection, InSb is a preferred semiconductor in embodiments of the present invention, and consequently the following discussion refers to InSb as an example. It is understood, however, that reference to InSb is part of an illustrative example and that such reference to a specific material is not exclusive of other semiconductors.

[0144] InSb has a very high electron mobility (the effective electron mass is very small compared to the electron mass) while the corresponding heavy hole mobility is about average (the effective hole mass is on the general order of the electron mass). Consequently, the emitter is preferably a highly doped n* region, which will have a large supply of electrons that will be diffused/injected into the gap.

[0145] The gap region is very interesting in this type of scheme. The gap region may be intrinsic to enhance the electron mobility and perhaps to make a good analog to the vacuum thermionic system. This approach will lead to the presence of unscreened electrons in the gap region, which will produce an opposing potential that will be greater when more electrons are emitted. In essence, unscreened carrier injection into the gap would be expected to lead to a Debye screening problem, and intuition suggests that this would lead to a reduction in current flow. If so, p-type doping for the gap region may be desirable in order to provide background holes to screen the injected charge. To study this, the following one dimensional numerical model is proposed.

[0146] In the discussion of analytic modeling in the last section, the barrier height is taken as an independent parameter. In the kinds of structure considered here and in the following discussion, the barrier height is determined by the relative doping of the different regions. Control over the barrier height is implemented ultimately through the selection of the doping level of the gap region. The use of a p-type region leads to quite a large barrier relative to the barriers which gave the best results as discussed previously. Consequently, using an n-type gap region is utilized in order to fix the barrier height to be close to optimum. Charge injection from a hot n* emitter region into an n-type gap region is not a standard problem in semiconductor device design. Therefore, a numerical model is proposed that will aid in the analysis of such problems.

[0147] a. The static thermal semiconductor equations in one dimension

[0148] To proceed, the static thermal semiconductor equations are required in one space dimension. These are:

[0149] Gauss's law: $\begin{array}{cc}\frac{d}{dx}\left(ϵE\right)=\rho & \left(2.1\right)\end{array}$

[0150] Charge density constitutive laws:

[0151] ρ=q(p−n+N_d⁺−N_a⁻)  (2.2)

[0152] Current continuity equations: $\begin{array}{cc}-\frac{1}{q}\frac{d}{dx}{J}_n=G_n-U_n& \left(2.3\right)\\ \frac{1}{q}\frac{d}{dx}{J}_p=G_p-U_p& \left(2.4\right)\end{array}$

[0153] Current constitutive laws: $\begin{array}{cc}{J}_n=q{\mu }_e\mathrm{nE}+{\mathrm{qD}}_e\frac{dn}{dx}-q{\mu }_{e}n{\alpha }_n\frac{dT}{dx}& \left(2.5\right)\\ J_p=q{\mu }_h\mathrm{pE}-{\mathrm{qD}}_h\frac{dp}{dx}-q{\mu }_{h}p{\alpha }_p\frac{dT}{dx}& \left(2.6\right)\end{array}$

[0154] Conservation of thermal energy: $\begin{array}{cc}\frac{d}{dx}Q-\left(J_n+J_p\right)E=S& \left(2.7\right)\end{array}$

[0155] The (J_n+J_p)E term enters this equation with a minus sign since the work done against the field goes into work done on the load resistance. The net result is that energy is taken out of the thermal degrees of freedom.

[0156] A glossary of conventional symbols used in semiconductor equations is given as follows:

[0157] ∈: Dielectric constant

[0158] E: Electric field

[0159] x: Position

[0160] ρ: Charge density

[0161] q: Magnitude of the electron charge

[0162] n: Electron concentration

[0163] p: Hole concentration

[0164] N_d⁺: Ionized donor concentration

[0165] N_a⁺: Acceptor ion concentration

[0166] N_d: Donor concentration in a gap region

[0167] N_d* : Donor concentration in an emitter region

[0168] J_n: Electron current

[0169] J_p: Hole current

[0170] G_n: Rate of generation of electrons per unit volume

[0171] G_p: Rate of generation of holes per unit volume

[0172] U_n: Rate of recombination of electrons per unit volume

[0173] U_p: Rate of recombination of holes per unit volume

[0174] μ_e: Electron mobility

[0175] μ_h: Hole mobility

[0176] D_e: Diffusion coefficient for electrons

[0177] D_h: Diffusion coefficient for holes

[0178] α_n: Electron thermopower

[0179] α_p: Hole thermopower

[0180] Q: Thermal heat flux

[0181] S: Thermal heat source

[0182] b. Zero-current conditions

[0183] In the absence of current flow, the semiconductor electron and hole concentrations will take on local thermal equilibrium values. The equilibrium relations, Equations (2.5)-(2.6), for the carrier concentrations can be found directly from a solution of the current constitutive relations with appropriate functional dependencies of the mobility, diffusivity and thermopower.

[0184] The following assumes that Maxwellian distributions are appropriate: $\begin{array}{cc}n=n_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}& \left(2.8\right)\\ p=n_i\left(T\right)e^{\frac{-q\phi }{k_{B}T}}& \left(2.9\right)\end{array}$

[0185] where n_i(T) is the intrinsic carrier density, and where ^φ is now the thermodynamic potential energy. This model presumes the presence of parabolic bands. The intrinsic carrier density is given by $\begin{array}{cc}{n}_i\left(T\right)=\sqrt{N_C\left(T\right)N_V\left(T\right)}{e}^{\frac{-E_g}{2k_{B}T}}\sim T^{\frac{3}{2}}{e}^{\frac{-E_g}{2k_{B}T}}.& \left(2.10\right)\end{array}$

[0186] where N_Cis the effective density of states for the conduction band, and N_Vis the effective density of states for the valence band. In the case of InSb, n_i(T) is evaluated to be $\begin{array}{cc}{n}_i\left(T\right)=4.9\times {10}^{15}{\left(\frac{m_{\mathrm{de}}{m}_{\mathrm{dh}}}{m_0^2}\right)}^{\frac{3}{4}}{T}^{\frac{3}{2}}{e}^{\frac{-E_g}{2k_{B}T}}=1.0\times {10}^{14}{T}^{\frac{3}{2}}{e}^{\frac{-E_g}{2k_{B}T}}{\mathrm{cm}}^{-3}.& \left(2.11\right)\end{array}$

[0187] where m_deis the effective electron diffusive mass and m_dhis the effective hole diffusive mass. The intrinsic carrier density for the relevant temperature regime is plotted in FIG. 6.

[0188] The conduction band in the case of InSb is known not to be parabolic. The dispersion relation of the valence band is given by (h²k²)/(2m₀*)=E(1+E/E_g), where m₀* is the effective mass at the minimum energy point of the conduction band. In the gap region, the electron concentration will be low and the electrons will be thermal, so that a parabolic approximation may provide a reasonable approximation. In the highly doped n* region, the effects of a non-parabolic band structure are expected to be seen. Relevant to the modeling, a reduction of the barrier height, relative to that computed from a parabolic model, is expected to be the primary impact.

[0189] C. The nonlinear Poisson equation

[0190] The equilibrium carrier densities can be utilized to develop a nonlinear Poisson equation for this model. Gaussφs law can be recast in terms of the electric potential ^φ: $\begin{array}{cc}\frac{d^2}{d{x}^2}\phi =\frac{q}{ϵ}\left[p-n+N_d^{+}-N_a^{-}\right]& \left(2.12\right)\end{array}$

[0191] where $E=-\frac{d\phi }{dx}.$

[0192] In the case of zero current flow, it is convenient to reference the electric potential to the thermodynamic potential, in which case the carrier concentrations become $\begin{array}{cc}n=n_i\left(T\right)e^{\frac{q\phi }{k_{B}T}},\mathrm{and}& \left(2.8\right)\\ p=n_i\left(T\right)e^{\frac{-q\phi }{k_{B}T}}.& \left(2.9\right)\end{array}$

[0193] Combining these equations leads to the nonlinear Poisson equation $\begin{array}{cc}\frac{d^2}{d{x}^2}\phi =\frac{2{\mathrm{qn}}_i\left(T\right)}{ϵ}\left[\sinh \left(\frac{q\phi }{k_{B}T}\right)-\frac{N_d^{+}-N_a^{-}}{2n_i\left(T\right)}\right].& \left(2.13\right)\end{array}$

[0194] This is now a temperature-dependent version of the nonlinear Poisson equation that frequently appears in the analysis of diodes and bipolar transistors.

[0195] It is convenient to define normalized variables for the nonlinear Poisson equation so that we may obtain a normalized version of the equation. The normalized potential is defined as $u=\frac{q\phi }{k_B{T}_{\max }},$

[0196] where T_maxis the maximum temperature at the emitter. The nonlinear

[0197] Poisson equation in terms of u becomes $\begin{array}{cc}\frac{d^2}{d{x}^2}u=\frac{1}{{\lambda }_D^2\left(T_{\max }\right)}\left[{\tau }^{\frac{3}{2}}{e}^{\frac{u_g}{2}\left(1-\frac{1}{\tau }\right)}\sinh \left(\frac{u}{\tau }\right)-c\right].& \left(2.14\right)\end{array}$

[0198] where the temperature-dependent intrinsic Debye length λ_D(T) is defined through $\begin{array}{cc}\frac{1}{{\lambda }_D\left(T\right)}=\frac{2q^2{n}_i\left(T\right)}{ϵk_{B}T}.& \left(2.15\right)\end{array}$

[0199] The relative temperature τ is $\begin{array}{cc}\tau =\frac{T}{T_{\max }}.& \left(2.16\right)\end{array}$

[0200] The normalized dopant concentration is $\begin{array}{cc}c=\frac{N_d^{+}-N_a^{-}}{2n_i\left(T_{\max }\right)}.& \left(2.17\right)\end{array}$

[0201] The normalized gap energy is $\begin{array}{cc}{u}_g=\frac{E_g}{k_B{T}_{\max }}.& \left(2.18\right)\end{array}$

[0202] The temperature-dependent nonlinear Poisson equation is not difficult to solve numerically. Solutions are illustrated in FIG. 7 under conditions where the doping concentration is given by $\begin{array}{cc}{N}_d^{+}-N_a^{-}={10}^{20}{e}^{-{\left(\frac{x}{1\mu }\right)}^2}{\mathrm{cm}}^{-3}-N_a& \left(2.19\right)\end{array}$

[0203] with N_ataken to be 5×10¹⁸, 10¹⁷,0, and −10¹⁷in units of cm⁻³. In this case the temperature profile was taken to be linear across the thermal diode, with T=350 K at the emitter side at x=0, and T=300 K at the collector side at x=10 μ. The symbol “μ” is used herein to mean μm, or micrometer. One observes that the barrier height is determined by the doping concentration. In the case where there are 5×10¹⁸holes/cm⁻³in the gap region, the barrier height is over 17 k_BT_max. From the discussion given above, it is known that this barrier is sufficiently great that the associated efficiency will be degraded. A barrier on the order of 7 k_BT_maxis preferred. FIG. 7 illustrates that such a low barrier would correspond to n-type doping in the gap region. FIG. 7 shows the normalized potential versus x for model doping concentrations with a linear temperature gradient.

[0204] A symmetric npn structure might be provided with an imposed temperature gradient. Furthermore, the collector and emitter regions may somehow ideally be connected to a material with a very tiny thermally-induced EMF. As the Debye length is on the order of 10 Ångstroms away from the depletion layer, it is a good approximation to take the local normalized thermodynamic potential to be $\begin{array}{cc}u=\tau {\sinh }^{-1}\left[e^{\frac{-u_g}{2}\left(1-\frac{1}{\tau }\right)}{\tau }^{\frac{-3}{2}}c\right]\approx \tau \ln \left[2e^{\frac{-u_g}{2}\left(1-\frac{1}{\tau }\right)}{\tau }^{\frac{-3}{2}}c\right].& \left(2.20\right)\end{array}$

[0205] The logarithmic approximation given is useful for positive c. This can be used to estimate the potential barrier between the emitter and a p-type gap in terms of the relative doping concentrations, which results in $\begin{array}{cc}\Delta u=\ln \left[\frac{N_d^{+}{N}_a^{-}}{n_i^2\left(T_{\max }\right)}\right].& \left(2.21\right)\end{array}$

[0206] The barrier in embodiments of this invention is up to about 7, preferably in the range from about 1.5 to about 7, and more preferably from about 3 to about 7. Unfortunately, it is not possible to achieve this for InSb in the absence of an independent gap bias for a p-type gap region in the vicinity of room temperature. In the case of an n* emitter to n-type gap region, the barrier height is $\begin{array}{cc}\Delta u=\ln \left[\frac{N_d^{*}}{N_d}\right].& \left(2.22\right)\end{array}$

[0207] A normalized barrier height of 5-7 corresponds to a doping ratio of e⁵-e⁷, which evaluates numerically to 150-1100. If the n* region is doped to a level of 10²⁰cm⁻³, then the gap region doping should be in the range of 9×10¹⁶cm⁻³to 7×10¹⁷cm⁻³.

[0208] The EMF potentially available from this idealized structure can be estimated using this kind of approximation, which results in $\begin{array}{cc}\delta u\approx \ln \left[\frac{N_d^{+}}{n_i\left(T_{\max }\right)}\right]-{\tau }_{\min }\ln \left[\frac{N_d^{+}}{n_i\left(T_{\min }\right)}\right].& \left(2.23\right)\end{array}$

[0209] The corresponding EMF in terms of δV for Nd_d⁺ is illustrated in FIG. 8. FIG. 8 illustrates induced EMF between the emitter and collector for an idealized InSb symmetric structure with N_d⁺=10²⁰. The collector is assumed to be at room temperature. One observes that the EMF in this case is not very large. In fact, it is the same EMF that would be obtained if the semiconductor were used as a thermoelectric.

[0210] The situation when current flow is present is now considered. For simplicity, the gap region is assumed to be sufficiently narrow that recombination can be neglected. In this case, the lossless semiconductor equations are used, which are written in the form:

[0211] Gauss's law: $\begin{array}{cc}\frac{d^2}{d{x}^2}\phi =-\frac{q}{ϵ}\left(p-n+N_d^{+}-N_a^{-}\right)& \left(2.24\right)\end{array}$

[0212] Current continuity equations: $\begin{array}{cc}\frac{d}{dx}{J}_n=0& \left(2.25\right)\\ \frac{d}{dx}{J}_p=0& \left(2.26\right)\end{array}$

[0213] Current constitutive laws: $\begin{array}{cc}{J}_n=-q{\mu }_{e}n\frac{d\phi }{dx}+{\mathrm{qD}}_e\frac{dn}{dx}-q{\mu }_{e}n{\alpha }_n\frac{dT}{dx}& \left(2.27\right)\\ J_p=-q{\mu }_{h}p\frac{d\phi }{dx}-{\mathrm{qD}}_h\frac{dp}{dx}-q{\mu }_{h}p{\alpha }_p\frac{dT}{dx}& \left(2.28\right)\end{array}$

[0214] Conservation of thermal energy: $\begin{array}{cc}\frac{d}{dx}Q-\left(J_n+J_p\right)E=S& \left(2.7\right)\end{array}$

[0215] To proceed, the current constitutive laws must be recast into a more convenient form. Recall that the carrier concentrations are taken at their equilibrium values when zero current flow is present. In light of this, the electron current constitutive law should be able to be written in the general form: $\begin{array}{cc}{J}_n=\left[\dots \right]\frac{d}{dx}\left[\frac{n}{n_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}\right].& \left(2.29\right)\end{array}$

[0216] In this form, if the electron concentration is locally in equilibrium, then the current density is identically zero. Comparing this form with the constitutive law given above results in $\begin{array}{cc}{J}_n={\mathrm{qD}}_e{n}_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}\frac{d}{dx}\left[\frac{n}{n_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}\right].& \left(2.30\right)\end{array}$

[0217] The analogous equation for the hole current is similarly $\begin{array}{cc}{J}_p=-{\mathrm{qD}}_h{n}_i\left(T\right)e^{\frac{-q\phi }{k_{B}T}}\frac{d}{dx}\left[\frac{n}{n_i\left(T\right)e^{-\frac{q\phi }{k_{B}T}}}\right].& \left(2.31\right)\end{array}$

[0218] These results can be used to compute electron and hole thermopowers as follows: $\begin{array}{cc}{\alpha }_n=-\frac{\phi }{T}+\frac{{\phi }_g}{2T}+\frac{3}{2}\frac{k_B}{q}& \left(2.32\right)\end{array}$

[0219] where E_g=q^φ_g. In the case of holes, the equation is $\begin{array}{cc}{\alpha }_p=-\frac{\phi }{T}-\frac{{\phi }_g}{2T}-\frac{3}{2}\frac{k_B}{q}.& \left(2.33\right)\end{array}$

[0220] FIG. 9 illustrates the computed thermopowers for electrons and holes for a thermal diode example with N_a⁻=5×10¹⁸.

[0221] Issues pertaining to energy conservation must be attended to. The first of these is that it is possible to connect the difference in the thermal heat flow to the load power at a fundamental level.

[0222] It is assumed that there are no energy sources within the semiconductor, so that $\begin{array}{cc}\frac{d}{dx}Q-J_{T}E=0& \left(2.34\right)\end{array}$

[0223] where J_T=J_n+J_p. After integration, the result is $\begin{array}{cc}{\int }_{x_o}^{x_n}\frac{dQ}{dx}dx-{\int }_{x_o}^{x_n}{J}_{T}Edx=0.& \left(2.35\right)\end{array}$

[0224] The computational boundaries are assumed at x_oon the emitter side, and x_non the collector side. The first integral is an integration over a derivative, which simply yields the difference in the heat flux at the two boundaries. The second integral can be analyzed using integration by parts $\begin{array}{cc}{\int }_{x_o}^{x_n}{J}_{T}Edx=-{\int }_{x_o}^{x_n}{J}_T\frac{d\phi }{dx}dx=-{\left[\phi \left(x\right)J_T\left(x\right)\right]}_{x_o}^{x_n}+{\int }_{x_o}^{x_n}\left[\frac{d{J}_T}{dx}\right]\phi dx.& \left(2.36\right)\end{array}$

[0225] The latter integral is zero because the total current is conserved. Consequently, it can be written

[0226] Q(x_n)−Q(x_o)+J_T[φ(x_n)−φ(x_o)]=0.  (2.37)

[0227] The loss in heat flux is balanced exactly by the product of the current density and the voltage drop, which is equivalent to the load power divided by the area. Whatever heat flux enters the device either emerges at the other end or else goes into driving the load. This is a particularly useful relation.

[0228] These results are also valid for incremental changes in x, so that the following constraint may be used equivalently $\begin{array}{cc}\frac{d}{dx}Q+\frac{d}{dx}\left(J_T\phi \right)=0.& \left(2.38\right)\end{array}$

[0229] The second important issue relating to energy conservation is the development of the heat flux in terms of the temperature gradient and the current densities. It is known that there are contributions to the total heat flux from lattice phonons, electrons and holes

[0230] Q=Q_L+Q_n+Q_p.  (2.39)

[0231] The lattice heat flux is given in terms of the lattice $\begin{array}{cc}{Q}_L=-{\kappa }_L\frac{dT}{dx}.& \left(2.40\right)\end{array}$

[0232] The lattice contribution to thermal conductivity κ_Lis available from the literature for InSb, and the data has been fitted to use in this model.

[0233] In the absence of current flow, there exists a carrier contribution to the thermal conductivity which is approximated by $\begin{array}{cc}{Q}_{n,p}={\kappa }_{n,p}\frac{dT}{dx}& \left(2.41\right)\end{array}$

[0234] where κ_n=Lqμ_enT and κ_p=Lqμ_hpT.

[0235] The parameter L in this equation is given (assuming acoustic deformation scattering) by $\begin{array}{cc}L={\left[\frac{k_B}{q}\right]}^2\left[2+{\left(4+\frac{E_G}{k_{B}T}\right)}^2\frac{{\mu }_e/{\mu }_h}{{\left(1+{\mu }_e/{\mu }_h\right)}^2}\right].& \left(2.42\right)\end{array}$

[0236] In the presence of current flow, the thermal flux should be modified since each carrier transports a thermal energy 3/2 k_BT. Including this leads to the overall result $\begin{array}{cc}Q=-\kappa \frac{dT}{dx}+\frac{3}{2}\frac{k_{B}T}{q}\left(J_p-J_n\right)& \left(2.43\right)\end{array}$

[0237] where the total thermal conductivity κ is made up of three contributions κ=κ_L+κ_n+κ_p. This leads to the modified energy conservation equation $\begin{array}{cc}-\frac{d}{dx}\left(\kappa \frac{dT}{dx}\right)+\frac{3}{2}\frac{d}{dx}\left[\frac{k_{B}T}{q}\left(J_p-J_n\right)\right]+\frac{d}{dx}\left(J_T\phi \right)=0.& \left(2.44\right)\end{array}$

[0238] d. Normalized equations

[0239] The normalized equations below are based on the normalizations given above, augmented with the normalized nonequilibrium carrier densities $\begin{array}{cc}\eta =\frac{n}{n_i\left(T_{\max }\right)},\mathrm{and}& \left(2.45\right)\\ \rho =\frac{p}{n_i\left(T_{\max }\right)}.& \left(2.46\right)\end{array}$

[0240] In terms of the normalized quantities, this leads to Gauss's law: $\begin{array}{cc}\frac{d^2}{d{x}^2}u=\frac{1}{{\lambda }_D^2\left(T_{\max }\right)}\left\{{\tau }^{\frac{3}{2}}{e}^{\frac{u_g}{2}\left(1-\frac{1}{\tau }\right)}\left[\frac{\eta -\rho }{2}\right]-c\right\}.& \left(2.47\right)\end{array}$

[0241] The current continuity equations will be directly implemented numerically, so that there is no need to normalize the current densities. The current constitutive laws can be written in terms of normalized carrier densities.

[0242] Current constitutive laws: $\begin{array}{cc}{J}_n={\mathrm{qD}}_e{n}_i\left(T_{\max }\right){\tau }^{\frac{3}{2}}{e}^{\frac{u_g}{2}\left(1-\frac{1}{\tau }\right)}{e}^u\frac{d}{dx}\left[\frac{\eta }{{\tau }^{\frac{3}{2}}{e}^{\frac{u_g}{2}\left(1-\frac{1}{\tau }\right)}{e}^u}\right],\mathrm{and}& \left(2.48\right)\\ J_p=-{\mathrm{qD}}_h{n}_i\left(T_{\max }\right){\tau }^{\frac{3}{2}}{e}^{\frac{u_g}{2}\left(1-\frac{1}{\tau }\right)}{e}^{-u}\frac{d}{dx}\left[\frac{\rho }{{\tau }^{\frac{3}{2}}{e}^{\frac{u_g}{2}\left(1-\frac{1}{\tau }\right)}{e}^{-u}}\right].& \left(2.49\right)\end{array}$

[0243] Conservation of thermal energy: $\begin{array}{cc}-\frac{d}{dx}\left(\kappa T_{\max }\frac{d\tau }{dx}\right)+\left[\frac{3k_B{T}_{\max }}{2q}\right]\frac{d}{dx}\left[\tau \left(J_p-J_n\right)\right]-\frac{d}{dx}\left(J_T\phi \right)=0.& \left(2.50\right)\end{array}$

[0244] e. Discretization

[0245] To develop numerical solutions for these equations a discretization scheme is required. The discretization of the semiconductor equations have been considered by many authors. At this point the numerical solution of the equations presents no particular difficulty. The only real questions have to do with whether to use uniform zoning or a nonuniform grid, and whether to go to the effort of developing a higher order discretization or not. For simplicity, a uniform grid is used with low order differencing. This has the advantage of reducing development time for the code. Further work in this area, if necessary, would presumably go toward implementing a numerical scheme based on a nonuniform grid.

[0246] The discretization of the nonlinear Poisson equation can be done trivially to O(h²) on a uniform grid using $\begin{array}{cc}\frac{u_{j+1}-2u_j+u_{j+1}}{h^2}=\frac{1}{{\lambda }_D^2\left(T_{\max }\right)}{\left\{{\tau }^{\frac{3}{2}}{e}^{\frac{u_g}{2}\left(1-\frac{1}{\tau }\right)}\left[\frac{\eta -\rho }{2}\right]-c\right\}}_j.& \left(2.51\right)\end{array}$

[0247] The finite difference approach in this case is equivalent to the use of finite elements.

[0248] The following uses a minor variation on the approach that is normally used on the current continuity equations. The basic idea is that the continuity equation in the absence of recombination implies for the electron current $\begin{array}{cc}{\left(J_n\right)}_{j+\frac{1}{2}}={\left(J_n\right)}_{j-\frac{1}{2}}.& \left(2.52\right)\end{array}$

[0249] Typically, the current density is eliminated in favor of carrier concentrations. It is well known that simply using the constitutive laws in this application leads to significant discretization errors since the carrier density can vary over several orders of magnitude. Instead, it has become customary to use integral relations to relate the current density to the carrier density. The basic approach, modified for these equations, involves starting with the constitutive law in the form $\begin{array}{cc}{J}_n={\mathrm{qD}}_e{n}_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}\frac{d}{dx}\left[\frac{n}{n_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}\right]& \left(2.30\right)\end{array}$

[0250] and then integrating to obtain $\begin{array}{cc}{\int }_{x_j}^{x_{j+1}}\frac{J_n}{{\mathrm{qD}}_e{n}_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}dx={\left[\frac{n}{n_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}\right]}_{x_j}^{x_{j+1}}.& \left(2.53\right)\end{array}$

[0251] In the absence of recombination, the current density is constant through the region, and we can solve for (J_n)_j+½ to obtain $\begin{array}{cc}{\left(J_n\right)}_{j+\frac{1}{2}}=\frac{{\left[\frac{n}{n_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}\right]}_{j+1}-{\left[\frac{n}{n_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}\right]}_j}{{\int }_{x_j}^{x_{j+1}}\frac{1}{{\mathrm{qD}}_e{n}_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}dx}.& \left(2.54\right)\end{array}$

[0252] The integral in the denominator can be approximated to low order using $\begin{array}{cc}{\int }_{x_j}^{x_{j+1}}\frac{1}{{\mathrm{qD}}_e{n}_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}dx\approx \frac{1}{{q\left(D_e\right)}_{j+\frac{1}{2}}{n}_i\left(T_{\max }\right){\tau }_{j+\frac{1}{2}}^{\frac{3}{2}}}{\int }_{x_j}^{x_{j+\frac{1}{2}}}{e}^{\left(\frac{u_g}{2}-u\right)/t}dx.& \left(2.55\right)\end{array}$

[0253] This presumes that the exponential factors give the fastest dependence on x to the integrand. Assuming that the argument of the exponential can be taken to be linear over the zone, the approximation obtained is ${\int }_{x_j}^{x_{j+1}}\frac{1}{{\mathrm{qD}}_e{n}_i\left(T\right)e^{\frac{q\phi }{k_{B}T}}}dx\approx \frac{h}{{q\left(D_e\right)}_{j+\frac{1}{2}}{n}_i\left(T_{\max }\right){\tau }_{j+\frac{1}{2}}^{\frac{3}{2}}}\frac{e^{\left(\frac{u_g}{2}-u_{j+1}\right)/{\tau }_{j+1}}-e^{\left(\frac{u_g}{2}-u_j\right)/{\tau }_j}}{\left(\frac{u_g}{2}-u_{j+1}\right)/{\tau }_{j+1}-\left(\frac{u_g}{2}-u_{j+1}\right)/{\tau }_j}.$

[0254] Using these results, the continuity equation becomes $\begin{array}{cc}\frac{{q\left(D_e\right)}_{j+\frac{1}{2}}{n}_i\left(T_{\max }\right){\tau }_{j+\frac{1}{2}}^{\frac{3}{2}}}{h}\frac{{\left[\frac{{\mathrm{\eta e}}^{-u/\tau }}{{\tau }^{\frac{3}{2}}{e}^{\left(\frac{u_g}{2}-u\right)/\tau }}\right]}_{j+1}-{\left[\frac{{\mathrm{\eta e}}^{-u/\tau }}{{\tau }^{\frac{3}{2}}{e}^{\left(\frac{u_g}{2}-u\right)/\tau }}\right]}_j}{\frac{e^{\left(\frac{u_g}{2}-u_{j+1}\right)/{\tau }_{j+1}}-e^{\left(\frac{u_g}{2}-u_j\right)/{\tau }_j}}{\left(\frac{u_g}{2}-u_{j+1}\right)/{\tau }_{j+1}-\left(\frac{u_g}{2}-u_{j+1}\right)/{\tau }_j}}=\frac{{q\left(D_e\right)}_{j-\frac{1}{2}}{n}_i\left(T_{\max }\right){\tau }_{j-\frac{1}{2}}^{\frac{3}{2}}}{h}\frac{{\left[\frac{{\mathrm{\eta e}}^{-u/\tau }}{{\tau }^{\frac{3}{2}}{e}^{\left(\frac{u_g}{2}-u\right)/\tau }}\right]}_j-{\left[\frac{{\mathrm{\eta e}}^{-u/\tau }}{{\tau }^{\frac{3}{2}}{e}^{\left(\frac{u_g}{2}-u\right)/\tau }}\right]}_{j-1}}{\frac{e^{\left(\frac{u_g}{2}-u_j\right)/{\tau }_j}-e^{\left(\frac{u_g}{2}-u_{j-1}\right)/{\tau }_{j-1}}}{\left(\frac{u_g}{2}-u_j\right)/{\tau }_j-\left(\frac{u_g}{2}-u_{j-1}\right)/{\tau }_{j-1}}}.& \left(2.56\right)\end{array}$

[0255] This equation is a numerically simple three-term relation for the normalized electron concentration η. A similar equation is obtained for the normalized hole concentration as follows: $\begin{array}{cc}\frac{{q\left(D_h\right)}_{j+\frac{1}{2}}{n}_i\left(T_{\max }\right){\tau }_{j+\frac{1}{2}}^{\frac{3}{2}}}{h}\frac{{\left[\frac{{\mathrm{\rho e}}^{-u/\tau }}{{\tau }^{\frac{3}{2}}{e}^{\left(\frac{u_g}{2}+u\right)/\tau }}\right]}_{j+1}-{\left[\frac{{\mathrm{\rho e}}^{-u/\tau }}{{\tau }^{\frac{3}{2}}{e}^{\left(\frac{u_g}{2}+u\right)/\tau }}\right]}_j}{\frac{e^{\left(\frac{u_g}{2}+u_{j+1}\right)/{\tau }_{j+1}}-e^{\left(\frac{u_g}{2}+u_j\right)/{\tau }_j}}{\left(\frac{u_g}{2}+u_{j+1}\right)/{\tau }_{j+1}-\left(\frac{u_g}{2}+u_{j+1}\right)/{\tau }_j}}=\frac{{q\left(D_h\right)}_{j-\frac{1}{2}}{n}_i\left(T_{\max }\right){\tau }_{j-\frac{1}{2}}^{\frac{3}{2}}}{h}\frac{{\left[\frac{{\mathrm{\rho e}}^{-u/\tau }}{{\tau }^{\frac{3}{2}}{e}^{\left(\frac{u_g}{2}+u\right)/\tau }}\right]}_j-{\left[\frac{{\mathrm{\rho e}}^{-u/\tau }}{{\tau }^{\frac{3}{2}}{e}^{\left(\frac{u_g}{2}+u\right)/\tau }}\right]}_{j-1}}{\frac{e^{\left(\frac{u_g}{2}+u_j\right)/{\tau }_j}-e^{\left(\frac{u_g}{2}+u_{j-1}\right)/{\tau }_{j-1}}}{\left(\frac{u_g}{2}+u_j\right)/{\tau }_j-\left(\frac{u_g}{2}+u_{j-1}\right)/{\tau }_{j-1}}}.& \left(2.57\right)\end{array}$

[0256] The energy equation, written in terms of the temperature, can be discretized without difficulty to give ${\kappa }_{j+\frac{1}{2}}\left({\tau }_{j+1}-{\tau }_j\right)-{\kappa }_{j-\frac{1}{2}}\left({\tau }_j-{\tau }_{j-1}\right)-\frac{3}{4}\frac{k_{B}h\left(J_p-J_n\right)}{q}\left({\tau }_{j+1}-{\tau }_{j-1}\right)-\frac{1}{2}\frac{k_B{\mathrm{hJ}}_T}{q}\left(u_{j+1}-u_{j-1}\right)=0$

[0257] The differential equation for the temperature equation (2.50) is seen to be a perfect differential of a lower order equation. It could reasonably be asked as to why not take advantage of this to reduce the equation to a first order equation, and develop a correspondingly simpler discretization scheme. It is somewhat more convenient to work with the second order system assuming that the temperatures are fixed at the boundaries. The second order system has two linearly independent solutions, which allows the specification of both of the boundary conditions. Working with a first order version of the equation can also handle two boundary conditions.

[0258] We should note that the energy equation described hereinabove does not conserve entropy. However, additional work using a more advanced model that incorporates entropy conservation gives similar results, with the efficiency numbers differing only by small factors from the corresponding efficiency numbers obtained with the foregoing formulae.

[0259] 3. Results for the InSb Thermal Diode

[0260] Analytic estimates have been described above for the solid state thermionic converter based on a semiconductor implementation, along with the presentation of a detailed numerical model that can be used for simulating such systems. This section presents the numerical results. There are numerous questions that might be asked. For example, the analytic model is based on the notion that carriers are injected from the emitter into the gap region. Can it be verified that the thermal semiconductor equations support such an effect using the numerical model? The analytic model predicts a current and voltage characteristic for the thermal energy converter that is similar to that of a diode. Does the numerical model also predict the same kind of current and voltage relation even qualitatively? In electronics applications, electrons injected into a p-type region diffuse as a minority carrier. In the present invention, it will prove to be convenient to inject electrons into an n-type gap region. The analytic model is based on diffusion of these excess carriers across the gap region. It is of interest to understand whether a diffusion model is relevant in this limit, and whether injected carriers can make it to the collector. Once these various issues are understood, there remains the question of the peak efficiency that such a system can obtain.

[0261] An issue that needs to be addressed concerns the specific emitter and collector configuration to be analyzed. While a symmetric emitter and collector configuration was analyzed above, the successful operation of such a design requires that the electron mobility be constant or increasing as a function of temperature. A realistic model for the mobility has a decreasing mobility as a function of temperature, which means that we must reduce the emission of the collector in order to get net thermionic energy conversion. The following discussion focuses on non-symmetric “diode”—like configurations, in which an emitter injects carriers into the gap as before, but where the collector injects electrons only at a level consistent with equilibrium carrier densities near the collector contact. If injection from the collector is significant, then it can be suppressed by reducing the supply of available carriers from which this current is derived.

[0262] Considered first are designs in which the carrier concentration at the collector is assumed to be in equilibrium, with no excess carrier injection assumed to occur from the metal contact. Such a model can be specified mathematically through the use of an equilibrium boundary condition at the collector boundary. Some thought must go into the physical implementation of such a boundary condition. An ohmic contact with a metal that has a large work function, and hence deeper occupied electronic states, would likely suppress collector emission. An alternative approach might be to create a relatively thin layer with a high recombination rate, which would force the carrier concentrations to take on equilibrium values. For example, if the collector side of the gap were damaged by rare gas ion bombardment prior to metallization, there would be a large concentration of defects that would produce a high recombination rate.

[0263] Referring again to FIG. 2, the emitter is the hot n* region 14 to the left. The gap region 16 is a thick region in the center, which may be either n-type or p-type (although it is found that the efficiency is higher if the gap is n-type). The collector is depicted here as a metal contact 20 that is cold.

[0264] The designs outlined above succeed in developing a significant thermionic energy conversion efficiency. This efficiency is ultimately limited by the presence of an ohmic return current due to the thermoelectric response of the semiconductor. If this return current could be suppressed, then a substantial increase in efficiency would be obtained. One scheme for doing so involves compensating the n-type substrate with p-type doping to produce a nearly intrinsic layer in front of the collector contact in embodiments of the present invention, thus reducing dramatically the supply of available electrons that would initiate an ohmic return current. This approach is considered in the following section. Within the numeric model, this approach appears to be successful, although a trade-off is apparent in that too much p-type compensation can restrict the flow of thermionic current to the collector. The calculations show that there is a small window within parameter space in which a compensated layer can be matched with the gap doping so as to simultaneously allow for nearly free thermionic current flow and nearly zero ohmic return current flow. The resulting designs have a very high predicted efficiency for energy conversion, and may be competitive with the best of the thermoelectrics. As there exists other semiconductors with better mobility to thermal conduction ratios (such as HgSe and HgTe), the compensated diode scheme may have the potential to develop conversion efficiencies well in excess of the best thermoelectrics. As will be shown hereinbelow,—see for example the discussion regarding FIG. 46—compensation layers developed in the context of this invention showed a significantly improved performance for materials such as InSb. This improvement was quantified in an increase in efficiency by a factor of about 2. For Hg_1−xCd_xTe, this improvement translates into efficiencies that approach the absolute limit of 50% of an ideal Carnot cycle efficiency.

[0265] Because these and other embodiments of compensated diodes, with such semiconductors that exhibit high mobility to thermal conduction ratios, can be designed by one with ordinary skill in the art in light of the teachings and examples provided herein, they are envisaged within the scope of the present invention.

[0266] A. Carrier injection

[0267] The premise of preferred designs of the present invention is that carriers are boiled off from a hot emitter region into the gap region, where they transport to the collector region. This section focuses on the issue of carrier injection from emitter into gap region, with three different choices of gap region type (p-type, intrinsic, and n-type). Electron injection into a p-type gap region would result in a much simpler problem to analyze, but there exists a significant barrier which occurs in the depletion region. The optimum efficiency occurs when the barrier is on the order of 4 k_BT. The barrier between an n* emitter and a p-type gap is closer to 8-9 k_BT. Consequently, to inject a larger number of carriers, a lower barrier is needed. Lower barriers occur with moderate n-type gap regions, but then one must understand carrier injection of a majority carrier.

[0268] In the case of thermal electron ejection into a p-type gap region, the analysis of the npn bipolar junction transistor can be used in order to have confidence that this is basically correct. The prior numerical model can also be used to examine this situation. FIG. 10 illustrates numerical solutions for charge emission from a hot n*-type emitter into a p-type gap region. It is seen that electrons are emitted into the gap region and screened by the majority carrier holes, and the minority carrier transport occurs primarily through diffusion. The holes act to screen the field in the gap region.

[0269] A significant reduction in the barrier height occurs when the gap region is intrinsic. As the injected carrier densities can be significant, a lightly doped n-type gap region is utilized with 10¹⁴cm⁻³donors that will simulate an intrinsic gap region. This simulated intrinsic gap region will have a lower potential barrier than a pure intrinsic gap region. Results for carrier injection in this case are illustrated in FIG. 10. FIG. 10 shows electron and hole concentrations for an n*pn* thermionic structure in InSb. The donor concentration in the emitter and collector regions is 10²⁰cm⁻³, and the acceptor concentration in the gap region is 10¹⁷cm⁻³. The emitter is at 600 K, and the collector is at 300 K. Three cases are illustrated: open circuit i=0 (equilibrium) with no carrier injection; short circuit v=0 with maximum current; and an intermediate case with v equal to half the open circuit voltage. It is seen that the electron and hole densities become very nearly equal, which implies that charge neutrality holds in the gap region to within what can be discerned on a graph of this type. It is thus verified numerically that charge injection into an intrinsic region is possible within the detailed model discussed in the last section. While it might have anticipated significant Debye screening and charge accumulation effects into the gap region, this seems not to occur. Electrons injected into the gap region appear to be able to transport to the collector region.

[0270] Results for carrier injection in this case are further illustrated in FIG. 11. FIG. 11 shows electron concentrations for an n*nn* thermionic structure in InSb. The donor concentration in the emitter and collector regions is 10²⁰cm⁻³, and the donor concentration in the gap region is 10¹⁴cm⁻³. The emitter is at 600 K, and the collector is at 300 K. Three cases are illustrated: open circuit i=0 (equilibrium) with no carrier injection; short circuit v=0 with maximum current; and an intermediate case with v equal to half the open circuit voltage. It is seen that the electron and hole densities become very nearly equal, which implies that charge neutrality holds in the gap region to within what can be discerned on a graph of this type. It is thus verified numerically that charge injection into an intrinsic region is possible within the detailed model discussed previously. While significant Debye screening and charge accumulation effects into the gap region might have been anticipated, this seems not to occur. Electrons injected into the gap region appear to be able to transport to the collector region without difficulty.

[0271] The case where carriers are injected from an n*-type emitter region into a moderately-doped n-type gap region is now examined. The numerical results are illustrated in FIG. 12. FIG. 12 shows electron and hole concentrations for an n*nn* thermionic structure in InSb. The donor concentration in the emitter and collector regions is 10²⁰cm⁻³, and the donor concentration in the gap region is 8×10¹⁷cm⁻³. The emitter is at 600 K, and the collector is at 300 K. Three cases are illustrated: open circuit i=0 (equilibrium) with no carrier injection; short circuit v with maximum current; and an intermediate case with v equal to half the open circuit voltage. It is observed that electron injection occurs, and that the transport still looks more or less diffusive.

[0272] The results presented above show that electron injection from an n* emitter region into the gap region occurs as expected in the case of a p-type gap, and also in both intrinsic and in n-type gap regions. Injection into a p-type gap region would be expected to follow a diode law. Heating the emitter region relative to the collector leads to a thermally-generated EMF. The collector in this configuration plays the same role as a metallized contact in a diode. Consequently, the current could be calculated directly from a diode law as was done in the analytic modeling. However, electron injection into intrinsic and n-type gap regions is now considered, for which there would be no reason a priori to expect a diode law to be satisfied. It is therefore of interest to investigate whether diode-type behavior extends into the new regimes, and what kinds of modifications might be expected.

[0273] To carry out such a study, current versus voltage characteristics are required, except that in the thermal energy converter there does not appear to be an adjustable voltage. The barrier height may be adjusted through a selection of the doping characteristic of the gap region. FIG. 13 illustrates the normalized barrier height for the example considered above as a function of the gap doping. FIG. 13 shows the normalized barrier height Δμ=qΔφ/k_BT_maxas a function of the doping concentration. The emitter is assumed to be doped to have 10²⁰electrons/cm³. The donors and acceptors in the gap are assumed to be completely ionized. Using this result as a mapping between doping and barrier height, the numerical model is used to simulate the short circuit current as a function of gap doping, and then create a plot of the magnitude of the current as a function of barrier height. The result is illustrated in FIG. 14. FIG. 14 shows short circuit current as a function of normalized barrier height for the structures considered above with T_max=600 K and Δτ=0.5. It is observed that the diode law is obeyed in the p-type regime away from intrinsic. It is also seen that, qualitatively, generic diode-type law behavior persists well into n-type regime. Such behavior is consistent with the postulated injection of carriers from emitter into gap region, with allowed transport across the gap to the collector. This provides a numerical verification that the general approach appears to be sound, and that the analytic modeling developed for this system based on diode-law behavior should be relevant.

[0274] B. The Current-Voltage Relation, Power and Efficiency

[0275] Having established the basic thermionic effect, attention now turns to the basic issue of the optimization of the efficiency. While it is possible to run the simulation for all relevant design parameters and carry out a purely numerical optimization, it may be useful to consider the systematic behavior of the solutions in order to understand what the numerical optimization is doing. For example, an examination of the current and voltage characteristic shows an approximate linear relation with offset, as might be expected from a voltage source and resistor. The optimization of the load power in this case is particularly simple. The thermal power that enters the device has a nearly constant offset due to the various Fick's law conduction terms, and an additional contribution due to the presence of current flow. Such a simple form for the thermal power allows for the development of a good approximate relation for the efficiency. An analytic optimization can be carried out, which is perhaps more illuminating as to what is going on in the device than the pure numerical optimization, which is in agreement with the analytic result.

[0276] The following begins with an examination of the numerical results for the current and voltage relation. Illustrated in FIG. 15 are representative curves that result from the numerical model. FIG. 15 shows current and voltage characteristic for the 625 μ thick InSb example considered above, with an emitter electron concentration of 10²⁰electrons/cm³, an emitter temperature T_max=600 K and a collector temperature T_min=300 K. Calculations are shown for gap donor densities of 10¹⁷(the lowest curve on the plot), 3×10¹⁷, 5×10¹⁷and 8×10¹⁷(the highest curve on the plot) in units of cm⁻³. It is shown that to within an excellent approximation, the current and voltage relations are linear. It is also noted that as defined above, electrons boiled off of the emitter will give rise to a negative current, as well as a negative open circuit voltage. FIG. 15 plots the magnitude of both current and voltage.

[0277] For the purpose of the optimization arguments that will follow, the current density as an arbitrary function of voltage is given approximately by $\begin{array}{cc}J\left(v\right)=J_s\left(1-\frac{v}{v_o}\right)& \left(3.1\right)\end{array}$

[0278] where J_sis the short circuit current density, and where v_ois the open circuit voltage.

[0279] The power delivered to the load per unit area is $\begin{array}{cc}{Q}_L=J\left(v\right)v=J_{s}v\left(1-\frac{v}{v_o}\right).& \left(3.2\right)\end{array}$

[0280] The current density in the computations is negative, corresponding to electron flow from left (emitter) to right (collector). The corresponding thermally induced voltage is positive. Consequently, we require J(v) and J_sto be defined here relative to the load, so that they are both positive when thermal power is being converted to electric load power. The numerical results of FIG. 16 agree well with this approximation. FIG. 16 shows the load power per unit area for the 625 μ thick InSb example considered above, with an emitter electron concentration 10²⁰electrons/cm³, an emitter temperature T_max=600 K and a collector temperature T_min=300 K. Calculations are shown for gap donor densities of 10¹⁷(the lowest curve on the plot), 3×10¹⁷, 5×10¹⁷and 7×10¹⁷(the highest curve on the plot) in units of cm⁻³.

[0281] The thermal power per unit area dissipated by the device for the conditions used in the previous examples is illustrated in FIG. 17. FIG. 17 shows the thermal power dissipated per unit area for the 625 μ thick InSb example considered above, with an emitter electron concentration 10²⁰electrons/cm³, an emitter temperature T_max=600 K and a collector temperature T_min=300 K. Calculations are shown for gap donor densities of 10¹⁷(the lowest curve on the plot), 3×10¹⁷, 5×10¹⁷and 7×10¹⁷(the highest curve on the plot) in units of cm⁻³. It is seen that the power is composed of a constant Fick's Law contribution at zero current (at the open circuit voltage), and a term linear in the current (and hence linear in the voltage). To within an excellent approximation, the dissipated power per unit area is $\begin{array}{cc}Q=Q_o+\frac{dQ}{dJ}J& \left(3.3\right)\end{array}$

[0282] where Q_ois the open circuit thermal power flow per unit area. Using the linear relation between voltage and current, this is equivalent to $\begin{array}{cc}Q=Q_s+\frac{dQ}{dv}v& \left(3.4\right)\end{array}$

[0283] where Q_sis the short circuit thermal power flow per unit area.

[0284] The efficiency of the solid state energy converter is then given approximately by $\begin{array}{cc}ϵ\left(v\right)=\frac{Q_L}{Q}=\frac{J_{s}v\left(1-\frac{v}{v_o}\right)}{Q_s+\frac{dQ}{dv}v}.& \left(3.5\right)\end{array}$

[0285] This is in reasonable agreement with the results illustrated in FIG. 18. FIG. 18 shows the efficiency as a function of voltage for the InSb example considered above. Calculations are shown for gap donor densities of 10¹⁷(the lowest curve on the plot), 3×10¹⁷, 5×10¹⁷and 7×10¹⁷(the highest curve on the plot) in units of cm⁻³. The dots are the optimum points as estimated by equation (3.8).

[0286] The approximate result for efficiency can be used in order to estimate the operating point that leads to optimum efficiency. For optimization, the point where the slope vanishes is solved for $\begin{array}{cc}\frac{d}{dx}ϵ\left(v\right)=0.& \left(3.6\right)\end{array}$

[0287] This leads to the result $\begin{array}{cc}{v}_{\mathrm{opt}}=\frac{1-\sqrt{\frac{Q_o}{Q_s}}}{1-\frac{Q_o}{Q_s}}{v}_o.& \left(3.7\right)\end{array}$

[0288] The efficiency at the optimum point is then $\begin{array}{cc}ϵ\left(v_{\mathrm{opt}}\right)=J_s{v_o\left[\frac{\sqrt{Q_s}-\sqrt{Q_o}}{Q_s-Q_o}\right]}^2.& \left(3.8\right)\end{array}$

[0289] Results from this analytic optimization are in reasonable agreement with the numerical optimization results illustrated in FIG. 18. FIG. 18 shows the efficiency as a function of voltage for the InSb example considered above. Calculations are shown for gap donor densities of 10¹⁷(the lowest curve on the plot), 3×10¹⁷, 5×10¹⁷and 7×10¹⁷(the highest curve on the plot) in units of cm⁻³. The dots in FIG. 18 are the optimum points as estimated by equation (3.8).

[0290] The results presented above show that the efficiency depends on the gap doping, as the gap doping determines the barrier height and corresponding current flow. A gap donor doping near 7×10¹⁷cm⁻³appears to be near the optimum. To explore this optimization further, FIG. 19 shows the results of a numerical optimization of efficiency as a function of gap doping over a wide range of doping densities at a fixed emitter ionized dopant concentration of 10²⁰cm⁻³. If the gap doping is near intrinsic, the efficiency is low. This is ultimately because the associated emitter-gap barrier is high. Accordingly, embodiments with near intrinsic gap doping that have a high emitter-gap barrier have a lower efficiency. If the gap doping gets to be too high, the beneficial effects of the low emitter-gap barrier is counter-balanced by the deleterious effect of the ohmic return current. This issue is examined further below.

[0291] C. Examination of scaling

[0292] The operation of the thermal diode model in this regime is primarily thermionic, and the scaling of the current, power per unit area, and efficiency follows qualitatively many features of the analytic model that we considered earlier. The following examines some scaling issues of the numerical solutions, essentially to verify the intuition developed from the analytic model.

[0293] The analytic model presumed a symmetric emitter and collector configuration (with temperature independent prefactors). The numerical model includes a more realistic mobility model, which in turn leads to a large return current from the collector. This means ultimately that the more accurate numerical model does not give efficiencies as large as what is suggested should be possible from the analytic model. For example, the χ parameter in the numerical computation is on the order of 0.07 in the gap region. From the analytic model, the optimum might be expected at a normalized barrier height φ˜4, with an associated efficiency in the range of 20%. The numerical results do lead to an optimum efficiency for φ→Δu˜4.5 which is obtained through optimization of the gap doping density. The associated efficiency at the optimum is only about 5.5%, which suggests that the prefactors have a strong effect on the optimized efficiency. This has been verified within the framework of the analytic model.

[0294] To achieve a larger efficiency, the return current must be reduced. The section below shows that it is possible to increase the efficiency by roughly a factor of 2 when return current is reduced.

[0295] The efficiency of the thermal diode design is a function of the emitter doping. The thickness is assumed to be 625 μ the gap doping is maintained at 7×10¹⁷cm⁻³, and the emitter and collector temperatures are assumed to be 600 K and 300 K, respectively. The results are shown in FIG. 20 for the optimum efficiency as a function of emitter doping with fixed gap doping N_D=7×10¹⁷cm⁻³. It is seen that the efficiency increases monotonically with the emitter electron concentration, but the scaling is less than linear. This is due to two interesting effects: the emitter-gap barrier increases at higher emitter doping since the gap doping was kept fixed; and the electron mobility decreases at higher carrier concentration. Both of these effects combine to reduce the beneficial impact of a larger emitter doping.

[0296] It is possible to implant Te (which is the lowest ionization energy donor) in the emitter at concentrations on the order of 10²⁰cm⁻³. Utilizing the TRIM-91 code, the results indicate that such a high dopant density will lead to the development of an amorphous emitter layer. Such a layer will have a different band gap, effective mass, and mobility than what we have modeled. In addition, one would expect that the recombination rate would be very high. Some consequences of this can be anticipated. Electron injection into the gap will be limited to emitter densities that are available on the order of one recombination length into the emitter as measured from the gap side. This will be the case down to spatial scales that are on the mean free path of the electron.

[0297] The consequence of this is that a large amorphous emitter region with a gentle scale length that is on the order of a micron will very likely present an effective doping density to the gap that is much less than the peak density achieved at the semiconductor edge. On the other hand, a sharp n* profile might allow free streaming of the electrons from the highly-doped amorphous region into a crystalline intermediate region. This latter picture would correspond more closely to the assumptions built into the model under discussion here.

[0298] Another important issue is that the ionization of the donor levels in the emitter will likely be incomplete. If the emitter region were crystalline at high doping levels, then the associated conduction band density of states would not be particularly large, so that the ionization balance of the donors will likely favor significant occupation of the donors. Data for the donor ionization energy is available (Te appears to have a donor ionization energy of 50 meV in InSb), so that the ionization fraction can be estimated. The use of a metal contact with a low work function at the emitter may circumvent associated problems, as the thermionic injection from a metal can be quite large.

[0299] The analytic model suggests that the optimum efficiency should be independent of the gap length. This is very nearly true of the numerical model for a gap thickness ranging between 200 μ and 2 mm. The thermal power is proportional to the inverse gap thickness in the analytic model, and also in the numeric model as illustrated in FIG. 21 for a thermal diode.

[0300] The previous discussion has focused on a total InSb wafer thickness of 625 μ. In the absence of recombination effects, thick layers would be preferred for large T_maxapplications, since the associated thermal flux would be correspondingly less. The recombination length of electrons in bulk n-type crystalline InSb for the gap densities that have been considered is at least 10 times the wafer thickness that has been examined. In addition, the total recombination rate is dominated by radiative recombination, which is likely to be strongly radiation trapped in large crystals such as considered here. Consequently, structures on the order of 1-2 mm should be interesting for energy conversion applications.

[0301] The effects of recombination on the device efficiency under conditions where recombination becomes important has not been addressed. While recombination effects might generally be assumed to be universally deleterious for thermionic energy conversion, this should probably not be second guessed in the absence of modeling results. The reason for this is that recombination increases the injected current over the diffusive limit considered here. While the thermal loss is inversely proportional to the gap length, the current would be inversely proportional to the recombination length. One might imagine that in this limit, the net result would be a net increase in the efficiency. This would be mitigated by the fact that the hole current required to support the recombination would come with an associated potential drop. As the hole mobility is low in InSb, the associated potential drop is likely to be large.

[0302] FIG. 22 shows the efficiency of a design with an emitter electron concentration of 10²⁰electrons/cm³and a gap donor density of 7×10¹⁷cm⁻³. The results are plotted as a function of the emitter temperature assuming that the collector temperature is 300 K. The results in the two cases are very nearly the same. The optimized efficiency as a fraction of the thermodynamic limit is illustrated in FIG. 23. One observes that the design works more or less equally as well compared to the thermodynamic limit at all temperatures shown.

[0303] When the temperature difference is lower, the thermal power flow is less for a given design. FIG. 24 shows the thermal power flow under conditions of optimum energy conversion for 626 μ and 1250 μ designs at different temperatures. The thermal power flow is in the range of a few hundred W/cm²under the optimum operating conditions of interest.

[0304] D. Summary

[0305] The previous discussion considered a model for a thermal “diode” based on an illustrative InSb implementation. This device uses a highly doped emitter region, a gap region which can be either p-type or n-type, and an ohmic metal collector with a sufficiently large work function arranged so as to have a negligible thermionic injection current above the ohmic contribution due to carrier equilibrium at the collector contact.

[0306] The results of the numerical and analytic studies reported above show that to obtain the largest thermionic injection current, an emitter to gap barrier is required that is preferably on the order of 4-5 k_BT_max, which implies that the gap needs to be preferably an n-type semiconductor. Therefore, this invention envisages embodiments whose emitter to gap barrier is preferably in the range from about 4 k_BT_maxto about 5 k_BT_max, as well as other embodiments in which this barrier is outside this range but that can be designed with the teachings provided herein.

[0307] As shown above, an n* region can inject electrons into an n-type gap region, and the transport is more or less diffusive in the gap region. Furthermore, it is also shown above that the thermal diode is capable of operation as an energy converter based on thermionic emission from the emitter into the gap, and subsequent transport to the collector. It has also been shown above how to optimize embodiments of the present invention as a function of gap donor concentration. The optimum efficiency of the thermal diode can be as high as 5.5% with a 600 K emitter, assuming that an electron density of 10²⁰cm⁻³can be developed in the emitter. These parameters are preferred characteristics of embodiments of this invention, although embodiments with other characteristics that depart from this particular optimization and which are designed according to the teachings provided herein are also envisaged within the scope of this invention.

[0308] 4. Results for the InSb Compensated Thermal Diode

[0309] A basic thermal diode structure implementation in InSb was considered as an energy converter. The efficiency of the device was found to be limited to somewhat over 10% of the thermodynamic limit. The analytic model considered above predicts a higher efficiency, and the difference between the actual experimental value and the analytic model prediction can be attributed to the temperature dependence associated with the electron mobility. Because of this dependency, the prefactors in the analytic model lead to a reduction in the maximum efficiency that can be obtained. The observation of this discrepancy and its cause suggests a compensation technique. A preferred compensation is implemented as follows.

[0310] This reduction of efficiency appears in the numerical models as an ohmic return current with a sign opposite to that of the thermionic current. It was noted in the previous section that it might be possible to increase the efficiency by suppressing this return current. One way to do this is to use p-type doping to produce a compensated layer on the inside of the collector contact, which would prevent injection of electrons from the collector side of the device (see FIG. 25). FIG. 25 schematically shows an embodiment of a compensated thermal diode according to this invention. The emitter is the hot n* region 14 to the left. The gap region 16 is a thick region in the center, which is n-type. The collector is depicted here as a metal contact 20 that is cold. The hot ohmic contact 12 is adjacent to the hot n* region 14. On the inside of the metal contact is a compensated region 19 created through the addition of p-type doping that suppresses the electronic return current. The addition of p-type doping can produce a layer of p-type semiconductor if not precisely matched to the substrate doping, which can inhibit the thermionic electron current from reaching the collector. A small region is found to exist in parameter space around perfect compensation in which design numbers can be chosen that allow simultaneous collection of the thermionic current and rejection of the ohmic return current. The efficiency computed for such a device was found to be substantially increased over that of the basic diode structure. This section considers this device and associated issues.

[0311] a. The return current

[0312] Before proceeding, the existence of the return current needs to be established in the basic thermal diode. One way to do this is to plot the current, including the sign, as a function of the gap donor concentration. The result is illustrated in FIG. 26. FIG. 26 shows the current as a function of gap doping for the illustrative InSb thermal diode design considered in the last section. For the thermionic regime, which corresponds to a negative current (electrons moving from left to right), the current is computed under conditions of optimum efficiency. For the thermoelectric regime, which corresponds to positive current (electrons moving from the collector to the emitter), the current is taken under conditions where the voltage is half of the thermally induced EMF, which is roughly where the optimum is in the thermionic regime. The thermally induced current is observed to change sign. At low gap doping, there is not enough conductivity for the gap to sustain an ohmic component large enough to compete with the thermionic injection. When the gap doping increases, at some point the ohmic current surpasses the thermionic current in magnitude, and thermionic energy conversion is no longer possible. This ohmic return current is made up of electrons that originate on the collector side of the device, and transport primarily by drift to the emitter.

[0313] b. Optimization of the efficiency with a blocking layer

[0314] We now consider the predicted operation of the compensated thermal diode, which includes an intrinsic blocking layer. In the simulation, we model the compensation layer using Gaussian p-type doping with a characteristic length of 5 μ. Results for different acceptor densities are shown in FIG. 27. FIG. 27 shows optimization of efficiency as a function of gap doping with p-type compensation using N_a⁻ concentrations of 7×10¹⁷, 10¹⁸, 2×10¹⁸and 3×10¹⁸cm⁻³. The dotted line indicates the efficiency obtained in the absence of a blocking layer. The maximum efficiency is obtained when the acceptor concentration of the compensation layer is adjusted to match the substrate donor concentration. A substantial increase in the optimum efficiency is obtained over the uncompensated case.

[0315] The shape of the efficiency curves shown in FIG. 27 can be understood qualitatively from simple considerations. The efficiency is maximized under conditions where the blocking layer is intrinsic, which simultaneously allows transmission of the thermionic current from the emitter, while producing a minimal return current. At lower gap donor concentration, the compensation layer produces a p-type region, which in this application behaves more or less as a reverse biased diode in rejecting the thermionic current. At higher gap donor concentration, the compensation is insufficient to eliminate excess electrons. A return current is initiated, with a magnitude that is roughly linear in the electron concentration in the blocking layer. Consequently, a linear decrease is seen in the efficiency on the high side of the optimum.

[0316] From a practical standpoint, the capacity of the electrical current leads limits the current densities to 10²-10³A/cm². Otherwise, the voltage drop in the wires becomes unacceptable. Moreover, there is a temperature drop across the diode of 200-300° C. For the given thermal conductivity of InSb, this translates into a gap thickness of about 1 cm. This thickness presents challenges such as the recombination length being comparable with the gap thickness and technological problems with polishing thick wafers. For example, most wafer processing equipment is designed for a thickness less than 1 mm.

[0317] A typical approach to achieving thick gaps is to stack the diodes. Because the current through the diodes stacked in series is the same, this means that stacked diodes should be current matched. One diode producing a larger current results in a voltage drop across the other diodes and reduced performance due to additional potential barriers.

[0318] The following discusses approaches to achieving current matching that can be implemented in the context of the present invention.

[0319] (1) Gap Doping Concentration Current Matching

[0320] The following example assumes an InSb diode material with all diodes having the same geometry and with a heat source temperature of 530 K and a heat sink temperature of 460 K. A single diode configuration is shown in FIG. 28. The arrow in FIG. 28 indicates that the temperature T of the hot ohmic contact 12 is higher than the temperature of collector 20. From the plot illustrated in FIG. 32 and setting an efficiency level at 6%, four diodes may be stacked as shown in FIG. 29, where the first diode (D_l) has a gap dopant concentration of 5·10¹⁷cm⁻³, D₂˜7·10¹⁷cm⁻³, D₃˜10¹⁸cm⁻³, and D₄˜2·10¹⁸cm⁻³. In this embodiment all four diodes are producing the same current, and the overall efficiency stays at 6%. Although some embodiments of stacked diodes according to the present invention comprise diodes whose respective elements are manufactured with the same materials for each diode, it is understood that embodiments of stacked diodes in the context of this invention are not limited to such stacks. Some embodiments of stacked diodes according to the present invention comprise diodes whose respective elements are manufactured with different materials. For example, in some embodiments of stacked diodes the emitters in different diodes comprise different materials, and/or the gap regions in different stacked diodes comprise different materials, and/or the collectors in different stacked diodes comprise different materials.

[0321] (2) Current Matching by Geometry

[0322] The following example assumes the highest efficiency line on FIG. 32, corresponding to a gap donor concentration of 2·10¹⁸cm⁻³. A thin diode in a series stack at 500 K produces a current that is approximately two times greater than the current produced in a non-stacked diode. Current matching will be achieved if the diode is at a temperature of 400 K and has an area that is twice as large as the area of a non-stacked diode. The results of this embodiment are shown in FIG. 30, where the area change is in the form of a wedge-shaped collection of diodes. The boundary of a stack is actually not a straight line, but rather a curve that takes into account non-linear behavior of the efficiency curve on FIG. 32. The stack shown in FIG. 31 shows another embodiment, where the stack boundary approximates an ideal curve as shown in FIG. 30. The arrows in FIGS. 30-31 indicate increasing temperature, T, from T_COLDto T_HOT.

[0323] The compensated thermal diode design has been optimized to operate with maximum efficiency with a hot emitter at 600 K. It is of interest to determine the efficiency of the device at other emitter temperatures. Numerical results for the efficiency are illustrated in FIG. 32 for different substrate dopings assuming perfect compensation. FIG. 32 shows the efficiency under optimized conditions as a function of emitter temperature for different gap doping with perfect compensation. An InSb compensated thermal diode structure 625 μ thick is assumed, with an emitter electron density of 10²⁰cm⁻³and a collector temperature of 300 K. Gap donor concentrations and matched acceptor concentrations are (in order of increasing efficiency as plotted) 7×10¹⁷, 10¹⁸, 2×10¹⁸and 3×10¹⁸cm⁻³. The dotted line indicates the efficiency obtained in the absence of a blocking layer. Results for the efficiency normalized to the thermodynamic limit are shown in FIG. 33. FIG. 33 shows the efficiency normalized to the thermodynamic limit under optimized conditions as a function of emitter temperature for the different cases shown in FIG. 32. One observes that the compensation layer is effective at high emitter temperature. In addition, optimization at high temperature appears to produce relative optima at other temperatures such that separate designs optimized for different temperature regimes is not required. More advanced designs that work best around their design temperature, and not as well at other temperatures, will be discussed below.

[0324] c. Example

[0325] A compensated layer in an n-type semiconductor can be made by, including but not limited to, introducing acceptors. For InSb doped with Te (donor impurity), the donor ionization energy is 50 meV. The same ionization energy is characteristic for acceptors created by vacancies. A compensated layer exists if the number of vacancies matches the initial donor concentration (n).

[0326] The number of vacancies is defined by the ion dose per unit area D (ions/cm²) and the number of vacancies created by a single ion V, if vacancies are induced by ion implantation of an inert gas. V is a function of the ion energy E, V=V(E). The number of vacancies is N_v(E)=D·V(E). In this case, the vacancy concentration c depends on the ion range in the material R, which is also a function of the ion energy R=R(E). Thus, for a given ion type, the vacancy concentration as a function of the ion energy is $C\left(E\right)=\frac{D·V\left(E\right)}{R\left(E\right)},$

[0327] for a compensated layer C(E)=n, or $\begin{array}{cc}D=\frac{n·R\left(E\right)}{V\left(E\right)}.& \left(4.1\right)\end{array}$

[0328] R(E) and V(E) were modeled using TRIM-91 computer code for InSb and Ne, Ar and Xe ions. The modeling results are shown in FIGS. 34 and 35. The dose needed to create a compensated layer over a wide range of ion energies in an n-type InSb diode doped to a concentration n can be determined utilizing FIGS. 34 and 35 and Equation (4.1). Lines 32 and 38 represent the curves for Argon, lines 34 and 36 for Neon, and lines 30 and 40 for Xenon in FIGS. 34 and 35, respectively.

[0329] Ion implantation creates a vacancy concentration profile which is more pronounced at the last 20-30% of the ion range. This 20-30% of the ion range can be decreased to less than the tunneling distance in InSb, which is typically between 100-150 Å, to avoid the formation of additional barriers.

[0330] Which ion is utilized requires a balancing of the pros and cons, such as the number of shallow levels that are created versus the damage to the solid structure. For example, Xe creates more shallow layers; however, it inflicts more damage and the crystal is semi-amorphous.

[0331] d. Summary

[0332] A modification of the basic thermal diode design introduced in the last section has been set forth which seeks to reduce the ohmic return current that limits the maximum efficiency of the device. It was demonstrated that an ohmic return current is present in the computations, and that it dominates the thermionic current at high gap doping. Including a compensated layer on the inside of the collector is shown to reduce the return current, leading to a higher optimum efficiency. The results of the modeling indicate that perfect compensation produces the highest efficiencies for a given gap donor concentration. This conclusion is consistent with what could have been reasoned a priori from simple physical arguments. Therefore preferred embodiments of compensated thermal diodes according to the present invention are characterized by high efficiencies, such that a larger compensation extent leads to a higher efficiency. The peak efficiency computed for the compensated thermal diode is competitive with the best thermoelectrics.

[0333] 5. Ohmic Contact

[0334] An ohmic contact is defined as a metal-semiconductor contact that has a negligible contact resistance relative to the bulk or spreading resistance of the semiconductor. (See Sze, S. M., Physics of Semiconductor Devices.N.Y., John Wiley & Sons, 1981, pp. 304-311, the contents of which are specifically incorporated herein.) This section describes preferred ohmic contacts and preferred methods for making such contacts according to this invention.

[0335] Metal-semiconductor interfaces introduce local potential barriers, which are known under the generic name of Schottky barriers. In its simplified form, the height of a Schottky barrier φ_bmeasured relative to the Fermi level can be written as φ_b=φ_m−χ_s, where φ_mis the metal electron work function and χ_sis the semiconductor electron affinity. Examples of the Schottky barrier values are 0.70 eV for GaAs and 0.18 eV for InSb.

[0336] For a solid state metal-semiconductor thermionic converter, the operating voltage range is lower than the Schottky barrier height. This will destroy the effect or at least bring down operating currents.

[0337] The typical operating voltage of the present invention is 10-100 mV and the power is 1-10 W. This results in an operating current of I_o=100-1000 A. The power loss on a Schottky barrier is W_loss=I_oφ_b. For W_lossto be less than 1% of the total power, φ_bmust be less than 1 meV. The barrier is often expressed in terms of a contact resistance. Therefore, at the stated currents, the contact resistance must be less than 10⁻⁵−10⁻⁶ohm.

[0338] The references Chang et. al., Specific Contact Resistance of Metal-Semiconductor Barriers. Solid-State Electronics,Vol. 14 (1971), pp. 541-550, and Shannon, J. M., Control of Schottky Barrier Height Using Highly Doped Surface Layers. Solid-State Electronics.Vol. 19 (1976), pp. 537-543, which are incorporated herein by reference, set forth a method of forming an ohmic contact. An electric field at the metal-semiconductor interface creates a carrier depleted region in the semiconductor. As the concentration of ionized impurity increases, the depletion width becomes narrower. This in turn causes the transmission coefficient for tunneling to increase. Hence, even a high barrier contact can become ohmic if the barrier is thin enough such that tunneling dominates the carrier transport process. A dopant level corresponding to 10⁻⁶ohm/cm²is 10²⁰-10²¹(Te in InSb) at 300° C. The electron effective mass for tunneling increases with temperature, and at 500° C. the required concentration is 10²¹rather than 10²⁰. A high dopant concentration layer must be sufficiently thin so it does not introduce its own barrier on the contacting semiconductor interface. The Shannon reference cited above estimates this thickness as less than 150 Å. This approach applies to both n-type and p-type doping, while keeping in mind that the current sign is reversed when going from an n-type region to a p-type region.

[0339] The implantation dose required to achieve a 10²¹cm⁻³shallow doping was calculated using TRIM-91 computer code (G. Ziegler, G. Biersack. IBM (1991)). The ion range and required dose were calculated separately for In and Sb. The calculation results were averaged to approximate InSb. The difference between In and Sb in this energy range was no more than 20%. Te was utilized as an n-type dopant because Te has the lowest known ionization energy (50 meV). FIG. 36 shows the calculation results for this dose, while FIG. 37 shows the ion range.

[0340] For p-type doping, the known materials for InSb include Ge (9 meV ionization energy) and Ag (E_i=30 meV). Ag is evidently a preferred ion since it is heavier than Ge and has a shorter range for the same ion energies, which allows for a smaller depleted region width. The calculations for Ag doping are shown in FIGS. 38 and 39. The ion implantation process creates vacancies which subsequently must be annealed.

[0341] Another method for forming an ohmic contact is through diffusion annealing. For example, an ohmic contact for a diode comprising InSb may be formed by annealing indium layers on InSb wafers. The following procedure was performed in an acid cleaned quartz ampule. The ampule was baked for more than one hour in a high vacuum at 800° C. The InSb samples having an indium coating were loaded into the quartz ampules, which was pumped down and filled with 10-100 torr of helium. Helium, which has a high thermal conductivity, provides for quick cooling. After annealing at various temperatures, I vs. V curves were measured on the samples to confirm that ohmic contacts existed. Positive results were obtained in the temperature range of250-400° C. with an annealing time of 10-60 minutes. At temperatures exceeding 500° C. the indium dissolved completely rendering the samples unusable, even though the samples showed ohmic behavior.

[0342] 6. Examples

[0343] a. Design parameters

[0344] Referring to FIG. 2, intermediate, thermally conducting layers may be placed in other embodiments of this invention between the ohmic contacts (12, 20) and heat sinks to ensure thermal contact. For example, a deposited layer of In or the like may be used on the hot side and a deposited layer of In-Ga eutectic or the like may be used on the cold side. These materials are sufficiently malleable to ensure adequate thermal contact at low compression (0.1-1.0 MPa).

[0345] Accordingly, preferred materials that can be used for these layers according to this invention are malleable thermal conductors, although other materials can be used in other embodiments. Another method of providing thermal contact is the application of paste, glue, a low temperature soldering alloy, or equivalents thereof. An electrically and thermally conducting layer is then added to serve as a diffusion barrier between a thermally conducting layer and a semiconductor. In this embodiment, the thermally and electrically conducting layer is used as an emitter without an additional semiconductor emitting layer. The characteristics and functions of this layer preferably include, but are not limited to, the following: (1) conducts heat; (2) conducts electricity; (3) emits electrons; (4) creates a Schottky barrier at the metal-semiconductor interface; (5) creates a diffusion barrier; (6) prevents a chemical reaction of a semiconductor with a subsequent layer; (7) matches the thermal expansion of a semiconductor to prevent delamination; (8) is thermally stable within the range of operation of the thermal diode; and (9) has a high resistance to oxidation if not vacuum encapsulated or encased in an inert environment.

[0346] InSb, for example, has a thermal expansion coefficient of 5.2-5.4×10⁻⁶K⁻¹in the temperature range of 300-500 K. Other possible materials include, but are not limited to, Mo, Cr, W, Ta, Re, Os, Ir, lanthanoids and nickel alloys, Pt and soft metals such as In, Au, Cu or the like. From this list, Ta and the lanthanoids are prone to oxidation, and In has a low melting temperature.

[0347] Highly doped semiconductors and semi-metals may also be used. For example, a thin layer of Si has a sufficiently high thermal and electric conductivity. However, certain precautions should be observed, and, in particular, it should be noted that a large forbidden gap when compared to InSb ensures the information of an internal barrier which impedes current transport.

[0348] The thickness of preferred embodiments of the thermally and electrically conducting layer is defined dby the following design rules. The thermal conductivity is preferably higher than that of a semiconductor gap. With a gap thickness of 100-1000 microns, the thickness of the layer is preferably less than about a few microns since it will increase thermal losses. On the thin side, there are a few considerations that define the layer thickness. For example, metal layers are preferably thicker than the electron mean free path in order to maintain its bulk properties. Since the layer is in close proximity to another metal (intermediate layer), it can affect its Fermi level position and change the electron emission into the semiconductor. This effect is known to be significant at metal layer thicknesses below 1000 Å. This number is at least a few electron mean free path lengths and can be regarded as a low practical limit in order to avoid unnecessary complications. Similar thickness considerations apply to the semiconductor emitter region n*.

[0349] The most preferred situation for the emitter-gap interface is when the region has matched crystallography, i.e., when the emitter region is grown epitaxially on top of the gap region. For InSb this can be achieved by maintaining the deposition temperature above 150° C. (PVD). For other gap materials, such as Hg_1−xCd_xTe, the epitaxial growth is more complex. Scattering and decreased converter performance occurs when the emitter-gap interface is mismatched.