One of the major challenges in the field of energy saving is the creation of devices for converting low temperature heat into electricity. Currently, thermionic and thermoelectric energy converters are used for these purposes.
Thermionic energy converters are vacuum diodes with electrodes placed at a relatively great distance from each other, which is not less than a few microns . Operating temperature of these converters is in the range of 1500–2500°C due to the large cathode work function, which is about 4–5 eV.
Thermoelectric energy converters are solid-state semiconductor devices with p-n junctions [2, 3]. They can work in the temperature range from -100°C to +600°C, but have a relatively low efficiency due to the high thermal conductivity of the semiconductor as a result of the significant contribution of lattice component in the overall thermal conductivity of the material.
The aim of this project is to define the basic characteristics of vacuum tunnel diodes (VTD) to assess the possibility of their use as generators of electricity in the temperature range from 50°C to 300°C.
VTD like thermionic devices consist of two flat electrodes placed in vacuum and separated by a gap of about 1 nm. The cathode is made of metal, and the anode of the n-type semiconductor. The cathode work function is above the anode work function and the cathode temperature is above the anode temperature. Thanks to the tunneling effect, for the electrons emitted from the cathode, it is not necessary to overcome a potential barrier of several electron volts. As in the Schottky diodes , they must overcome the potential barrier of much smaller height, which is equal to the difference of electron work function for metal and semiconductor. Depending on the type of metal and semiconductor and on semiconductor doping, the difference between these works can be adjusted in the range from 0.1 eV to 1 eV, so the effective power generation can be achieved at much lower temperatures. In addition, the vacuum gap between the electrodes eliminates the lattice component of thermal conductivity, which allows to increase the efficiency of the vacuum tunnel diodes.
The calculation of the main characteristics of the VTDs, such as the electric current density, power density, electrical power and efficiency depending on voltage, was carried out under the following conditions:
• electrodes are flat and parallel to each other. The characteristics of VTDs depend only on the coordinate z in the direction perpendicular to the electrodes plane;
• VTD is the state of thermodynamic equilibrium;
• the electric current does not violate thermodynamic equilibrium;
• the absence of surface states in the semiconductor;
• reduction of the potential barrier height in a semiconductor due to image forces is not taken into account;
• the width of the depleted layer in the semiconductor is less than the mean free path of the electrons, so calculations of the density of the electron beam can be performed in a ballistic mode;
• no heat transfer caused by lattice thermal conductivity, due to the vacuum gap between the electrodes;
• temperature of the metal electrode is above the temperature of the semiconductor electrode (Tm > Tn).
Since the electron work function for metal is higher than for n-type semiconductor (Φm > Φn), electrons leave the semiconductor, leaving it positively charged, and charge metal electrode negatively.
With the above assumptions the following expression are valid for electric current density j and the energy flux density q carried by the electrons in the z axis direction in the ballistic regime [6, 7]:
where qe is the electron charge; m is the electron mass; h is Planck constant; Er is the energy of electrons with momenta lying in planes parallel to the electrode surface; Ez is electrons energy with momenta lying in the direction perpendicular to the surface of the electrode; F is the Fermi energy; Kb is Boltzmann constant; T is absolute temperature; Vd is the voltage at VTD; τ(Ez,Vd) is the transparency coefficient of a potential barrier.
To estimate the transparency coefficient for electron tunneling through a potential barrier τ(Ez,Vd), which form is represented in Fig.1, the following formula can be applied in quasiclassical approximation :
where the Ez is component of the electron energy in the direction perpendicular to the surface of the electrodes; z1 и z2 are points on the curve of the potential barrier with the same values of Ez; τo is the coefficient close to 1.
Fig.1 shows various shapes of the potential barrier in the semiconductor U(z, do, Vd) depending on the distance z for different values of the vacuum gap do. It follows that with a decreasing of do the electrons can effectively tunnel through the vacuum gap, and the height of the potential barrier at the semiconductor surface increases to a value equal to the difference between the electron work functions for metal and semiconductor.
Characteristics of VTDs were calculated for the following parameters:
• difference between the electron work function for metal and n-type semiconductor ΔU = 0.1–1.0 eV (the work function for silicon Φn = 4.2 eV);
• donor impurity concentration Nn = 1020–1025 m-3;
• distance between electrons do = 0.1–5.0 nm;
• temperature difference between hot metal and cold semiconductor electrodes ΔT = 0–300 K at a fixed temperature of the semiconductor electrode Tn = 300 K.
Fig.2 shows the dependences of electric current density j(do, Tm, Vd) and energy flux density q(do, Tm, Vd) on the voltage Vd on VTD for Tm = 500 K and for the difference of the work functions of 0.2. The dependence of electric current density j(do, Tm, Vd) on Vd is a voltage-current characteristic of the VTD, where the open-circuit voltage at j = 0 is equal to 0.07 V, and the short-circuit current at Vd = 0 is equal to 700 A/cm2. The maximum density of energy (heat) flux carried by electrons is 130 W/cm2 at the temperature difference between the electrodes of 200 K.
If the thickness of the electrodes is 0.2 mm, then for molybdenum with thermal conductivity of 131 W/m·K at temperature of 500 K and for silicon with thermal conductivity of 148 W/m·K at temperature of 300 K, when the energy flux is equal to 130 W/cm2, a temperature drop for metal electrode reaches 2 K, and for silicon electrode – 1.8 K. Thus, to create in vacuum gap the temperature difference of 200 K, the temperature difference between electrodes of VTD should be equal to 203.8 K.
Calculations show that with decreasing of Nn the maximum density of electric current and energy flux decreases.
The specific electrical power generated by the VTD is equal to the product of voltage and electric current density. The efficiency of the vacuum tunnel diode is defined as the ratio of the generated specific electrical power to the energy flux density transferred by electrons. Fig.3 shows typical dependences of the specific electric power P(do, Tm, Vd) and efficiency η(do, Tm, Vd) on the voltage Vd for Tm = 500 K and different values of the vacuum gap width do.
Fig.3 shows that the calculated dependences of specific electric power and efficiency on the voltage on VTD have bell-shaped form. Similar dependences of the efficiency are obtained for other concentrations of the donor impurity. With the decrease of Nn decreases specific electric power and increases efficiency.
Fig.4 presents the dependences of maximum specific electric power Pmax(do, Tm) on Tm for Tn = 300 K, Φm = 4.4 eV, Φn = 4.2 eV, do = 0.6 nm and the concentration of electrons, Nn = 1025 m–3. For comparison also the maximum values of the efficiency in the Carnot cycle ηcarno = (Tm–Tn)/Tm are shown.
Analysis of the calculation results allows to draw the following conclusions:
• electric power and efficiency increase with increasing temperature difference between the electrodes;
• generated electric power increases and the efficiency slightly decreases with increasing concentration of atoms of donor impurity;
• generated electric power decreases and efficiency increases with increasing difference between the work functions;
• generated electric power and the efficiency decrease with increasing width of the vacuum gap (stronger for small values of do < 0.4 nm and weaker for large values of do > 0.6 nm);
• efficiency is in the range of 40–50% of maximum possible value in the Carnot cycle.
From a practical point of view in the temperature range of 350–600 K, the concentration of electrons, Nn = 1025 m–3, the difference in the work functions of 0.2–0.3 eV and the width of vacuum gap of 0.5–0.8 nm are optimal, because at such values the maximum specific electrical power reaches up to 15 W/cm2 and an efficiency is in the range of 45–50% of maximum possible value in the Carnot cycle.
The calculated characteristics of VTD in the temperature range of 350–600 K exceed similar characteristics of thermoelectric energy converters. Thus, the specific generated power and efficiency of vacuum tunneling diodes are in 1.5–3 times and in 3 to 4 times higher than the corresponding characteristics of the thermoelectric energy converters. To obtain the required values of voltage and current of electric power generators separate VTDs can be connected in series/parallel.
The estimation of the characteristics of VTDs shows the prospects of their use as generators of electricity in the temperature range of 350–600 K.
One of the main challenges in creating VTDs is to develop a fundamentally new technology of manufacturing of structures with a vacuum gap of 0.5–0.8 nm, which is equal to two or three interatomic distances in solids. However, the development of nanotechnology in electronics and MEMS allows to hope that in the near future the difficulties will be overcome.
The deposition on the semiconductor of perforated one-triatomic layers of insulating materials, monomolecular layers of organic compounds and graphene films can be possible technological solution in the manufacturing of VTDs with a sub-nanometer vacuum gap.