Semiconductor Converters of Radiation Energy

V. S. Vavilov

1. Introduction

In the theory of physical processes in semiconductors and in the practical application of semiconductor devices, great advances have been achieved in recent years.

It has become quite evident that semiconductor devices are not only worthy competitors of vacuum electron tubes, but also open up new paths in radiophysics and electronics that had previously been closed.

Below we shall consider the possibilities that have opened up in a new field of application of semiconductors, connected with the problem of the rational utilization of the energy of solar radiation and the energy of radioactive decay.

It has been shown that light energy and the energy of radioactive decay can be directly converted, with considerable efficiency, into electrical energy by means of electron–hole (e–h) junctions in semiconductor crystals. Pairs of positive and negative charge carriers (holes and electrons), arising as a result of photon absorption or ionization inside the crystal during the passage of a fast electron and diffusing toward the junction, are separated in the field of the space charge of the junction. The carriers separated by the field change the height of the potential barrier, which leads to the passage of current through the load resistance connected between the external electrodes.

In view of the fact that each β-particle creates many pairs of charge carriers, an “element” consisting of a source of β-radiation and a semiconductor crystal with an e–h junction converts the flux of a small number of fast electrons into a considerably stronger current of low-energy electrons.

At present, other direct methods of converting the energy of radiation into electrical energy are also known. Academician A. F. Ioffe and his students developed semiconductor thermoelements, used with great efficiency[^2],[^24] in the national economy of the USSR. As early as 1913, Moseley[^3] established that a potential difference of 150,000 volts could be achieved by the emission of charged particles from a sphere containing radon and suspended on an insulator in a vacuum; in 1945 the Soviet physicists Lobanov and Belyakov reported results obtained with a high-voltage generator improved by them, based on the principle of charging a body by the emission of particles[^4]. Similar devices are also described in [^5] and [^7].

The phenomenon of the release of electrical energy as a result of illumination or irradiation of a rectifying layer by fast particles has long been known. The photoelement with a blocking layer was invented by the Russian physicist V. A. Ulyanin in 1888[^8]. However, the possibility of obtaining high conversion efficiencies[^9], which makes this phenomenon especially important, and the mechanism of the processes of the internal photoeffect near the region of the rectifying junction were clarified only recently in connection with general advances in the theory and applications of semiconductors[^1],[^10].

Of greatest interest are three areas of application of semiconductor “batteries.”

The first of these is compact, long-lived, and reliable power sources for electronic equipment with crystalline triodes. The creation of batteries of this kind with radioactive energy sources appears possible. An electrical power of the order of microwatts, with a potential difference of several tenths of a volt, can be achieved by using a single p–n junction and comparatively weak sources of fast electrons. Apparently, β-emitters of high specific activity with sufficiently long half-lives, which are products of the fission of heavy nuclei (Sr^90—Y^90), or which arise upon irradiation by slow neutrons in a reactor (for example, Tl^204), can be obtained at comparatively low cost.

A second field of application is the conversion of the decay energy of considerable quantities of unpurified or partially enriched fission products into electrical energy. In these cases the energy of both β- and γ-radiation can be used; devices for energy conversion require powerful protection from radiation and from large quantities of both fission products and semiconductor material.

The third, probably the most important field of application is the conversion of the energy of solar radiation.

2. ELECTRON–HOLE JUNCTIONS IN SINGLE CRYSTALS OF GERMANIUM AND SILICON

Electron–hole junctions, whose rectifying properties are used in semiconductor electronic devices (detectors, rectifiers, triodes, photoelements, etc.), exist in that part of a semiconductor crystal where the concentration of impurities changes more or less abruptly from a predominance of the acceptor type (the region of hole conductivity) to a predominance of the donor type (electronic conductivity)*). A large number of works by Soviet and foreign authors have been devoted to the theory of current rectification and the penetration of nonequilibrium charge carriers into a region with the opposite type of conductivity (see¹). In the present review we shall touch only upon those phenomena in e–h junctions which are directly necessary for considering the question of the conversion of radiation energy. In principle, an e–h junction should exist at the contact surface of two semiconductor crystals differing in type of conductivity. The presence of surface contamination, disturbance of the crystal-lattice structure at the surface of the crystal, and the difficulty of making a contact of considerable area do not make it possible to use this method for creating e–h junctions in a semiconductor. It has now been shown that near the point contact of metallic tips with a semiconductor, when the contact is formed by passing current pulses, rectifying e–h junctions arise. However, for the purposes of converting the energy of radiation absorbed by a semiconductor crystal, electron–hole junctions must have a considerable area, so that the greatest possible fraction of the excess charge carriers produced by absorption of radiation in the layer near the junction is captured by the electric field of the space charge of the junction.

Several methods are known for the artificial preparation of semiconductor crystals containing electron–hole junctions of considerable area. The properties of the junctions, i.e., the concentrations of donor and acceptor impurities on both sides of the junction and the steepness of the junction, determined by the gradient of impurity concentration, are controllable to a certain extent, i.e., they can be specified in advance. The bombardment of crystals with electronic conductivity by α-particles or deuterons¹,¹⁰, used in the first experiments to obtain e–h junctions, leads, along with the formation of acceptor centers, to a strong decrease in the diffusion length of the charge carriers; moreover, the method itself is complicated and is not used at present.

*) See, for example, the reviews in Uspekhi Fizicheskikh Nauk 40, 120 (1950), and also 46, 96 (1952).

The simplest from the technical standpoint is the so-called alloying-in method, widely used for producing rectifying e–h junctions in germanium rectifiers and triodes (see¹, articles 2, 3, 4). Indium, which is an acceptor impurity, is placed in the form of a small cylinder or plate on a single crystal of germanium with electronic conductivity; treatment at a temperature above the melting point of indium (155°) and below the melting point of germanium (936°), in vacuum or in an atmosphere of inert gas, leads to dissolution of part of the germanium in the liquid indium and to gradual advancement of the boundary Ge — alloy In + Ge into the crystal. A certain number of indium atoms diffuse from this boundary into the depth of the germanium, changing the type of conductivity. The area of the junction is limited, in principle, only by the dimensions of the indium plate and of the germanium crystal. A drawback of rectifying junctions obtained by the method of alloying in indium is the disturbance of the crystal structure in the region of hole conductivity, which is a thin layer between the germanium that has retained its original properties and the In + Ge alloy. The deterioration of the current-voltage characteristics is associated with disturbances of the structure.

Junctions with properties very close to those theoretically expected can be obtained by the method of thermal diffusion of impurities into single crystals of germanium and silicon. The diffusion constants of elements that are the most important acceptor and donor impurities in Ge and Si (Sb, As, Ga, In—for germanium; B, P, Li—for silicon) are known, and the realization of the conditions necessary for diffusion into the crystal of sufficient quantities of impurity atoms does not present significant difficulties. The crystals in which e–h junctions are to be produced are coated with a thin layer of the required impurity or are placed in an atmosphere of impurity vapor and heated at a sufficiently high temperature. The treatment time usually does not exceed tens of hours. The structure of single-crystalline germanium and silicon specimens, provided the necessary precautions are observed (temperature control during diffusion, slow cooling), remains unchanged. According to the results reported in⁹, in large-area silicon photocells the diffusion length of charge carriers reaches 0.1–0.2 mm, which is sufficient for obtaining high efficiencies in the conversion of the energy of light radiation.

Methods of introducing an impurity during the growth of Ge and Si single crystals, as well as of displacing impurities by changing the growth rate, have been developed to a lesser degree (see¹). The reason lies in the considerable complexity of carrying out experiments, which does not permit the production of large numbers of crystals with junctions, and also in the lesser definiteness of the geometry of the junctions obtained by the indicated methods.

Single crystals of germanium and silicon of high purity, most often used for the manufacture of semiconductor devices with p–n junctions, as well as crystals of intermetallic compounds of elements of groups III and V of the Mendeleev system, similar to Ge and Si in their electrophysical properties, have two bands of allowed electron-energy values separated by a forbidden band, whose width is of the order of one electron-volt. In Ge and Si crystals these are the valence band, almost filled with electrons, and the nearly empty conduction band lying above it. The width of the forbidden band in germanium is 0.75 eV, and in silicon 1.15 eV.

Figure 1 shows the energy levels in the region of a p–n junction in equilibrium. In the region of the junction

Fig. 1.

there is a double layer of charges and the electrostatic (contact) potential difference associated with it.

The height of the electrostatic barrier \(V_k\) is equal to the difference in the heights of the positions of the conduction band in substances with electron and hole conductivity.

The magnitude \(V_k\) depends on the concentration of charge carriers of the majority type in the electron and hole regions:

\[ V_k=\frac{kT}{q}\ln\frac{n_n p_p}{n_i^2}, \]

where \(k\) is Boltzmann’s constant, \(T\) is the absolute temperature, \(q\) is the electron charge, \(n_n\) is the concentration of electrons in the n-region, \(p_p\) is the concentration of holes in the p-region, \(n_i\) is the concentration of electrons or holes in a semiconductor without impurity levels, with intrinsic conductivity. For \(n_n\) and \(p_p\) of the order of \(10^{16}\ \mathrm{cm}^{-3}\), \(V_k\) is close to \(E_g\).

In the following sections an approximate theory of semiconductor energy converters with p–n junctions is considered,¹² along with the results of published experimental work.

On the basis of the theory of rectification in a p–n junction and an equivalent circuit, the maximum power and the efficiency of conversion are determined. The theory is applicable irrespective of the primary process of generation of excess charge carriers (see also²³).

3. CONVERTERS OF β-RADIATION ENERGY

The basic condition whose fulfillment is necessary in any semiconductor converter with a p–n junction is the absorption of as large a fraction as possible of the incident radiation near the junction.

The problem may be divided into: a) selection of the radiation source, b) choice of the geometry of the converter, and c) determination of the optimal properties of the semiconductor and the p–n junction.

It is obvious that radioisotopes emitting only β-particles are the most advantageous from the standpoint of the particle range, their mean energy, and the requirements for radiation shielding. Among the known, well-studied, and easily obtainable radioisotopes there are several isotopes with the required energies and half-lives. Absorption of γ-radiation by matter, as is known, is accompanied by the appearance of fast electrons, which in turn form pairs of charge carriers. Although the absorption coefficient of γ-rays is small, their use appears possible in devices of large dimensions.

B. M. Konovalenko, S. M. Ryvkin, and V. M. Tuchevich²² showed that the sensitivity of germanium photocells with p–n junctions to X-radiation exceeds by a factor of 100 and more the sensitivity of other types of photocells with a blocking layer.²² Germanium photocells in combination with portable pointer-type instruments constitute a very convenient and simple X-ray dosimeter. Alpha particles, passing through a semiconductor crystal, also give rise to the appearance of pairs of carriers, but their use is hardly possible because of the strong lattice damage arising in the substance. In the case of β-particles with energies exceeding a certain limiting value \(E_t\), accumulation of lattice defects occurs, causing a decrease in the conversion efficiency. One way out is to use sources with a limiting β-spectrum energy smaller than \(E_t\). In principle, in this case higher efficiencies may be obtained than when the mean energy is reduced by absorbers. In the theoretical treatment it is convenient to use a model of a plane layer of radioisotope on a crystal

of a semiconductor with a p–n junction parallel to the plane of the source and situated from it at a distance smaller than the diffusion length of the charge carriers. A device made of two semiconductor plates, one on each side of the radioisotope layer, is more effective.^12 Still more advantageous is a set of alternating parallel thin plates of semiconductor and active layers; in this case all the energy of the β-particles is utilized, and each p–n junction is bombarded from two sides.

The most important requirements on p–n junctions, from the standpoint of obtaining high efficiencies, are the following. The range of β-particles in the semiconductor must be small, while one of the diffusion lengths of the minority charge carriers (or their sum), which determines the effective working volume, must be large. In particular, it must substantially exceed the length over which β-particles with initial energy less than the threshold energy for the formation of lattice defects, \(E_t\), are absorbed. Let us recall that in pure germanium crystals the diffusion lengths of carriers sometimes reach several millimeters. The threshold energy for the formation of lattice defects, \(E_t\), should have as large a value as possible, while the energy \(\varepsilon\) expended by β-particles in forming a pair of carriers should be small. Some of the indicated properties are interrelated. If one of the diffusion lengths is small, as is the case in p–n junctions made by the method of alloying a metal into a semiconductor (see above), the junction must be located farther from the surface than the mean range of the β-particles, so that carrier liberation takes place in material with a large diffusion length. Since the width of the forbidden band \(E_g\) represents the upper limit of the potential difference at the output of the converter, it is desirable that it be large. A comparison of the properties of germanium and silicon indicates that silicon is more advantageous with respect to \(E_g\) (1.15 eV), whereas germanium has advantages in the range of β-particles, \(E_t\), and diffusion lengths.

The results achieved at present with silicon converters exceed, both in maximum voltage and in power, what has been obtained with germanium by at least an order of magnitude.

4. EQUIVALENT CIRCUIT AND EFFICIENCY OF THE CONVERTER

The potential difference, current, and electric power in the external circuit can be calculated by means of an equivalent circuit, on the basis of the current–voltage characteristic of the p–n junction. The maximum power delivered to the external circuit at the optimum load, and the limiting efficiency, are functions

of the flux of excess carriers and the differential resistance of the junction at zero displacement \(R_0\).

Excess charge carriers, arising at distances from the junction small in comparison with the diffusion length, under the assumptions of the theory being considered all penetrate through the junction, with holes going into the region with hole conductivity and electrons into the region with electron conductivity. Let \(I_g\) denote the current corresponding to the sum of the diffusion fluxes of holes and electrons through the junction. The diffusion flux of charge carriers through the \(p\)-\(n\) junction gives rise to a potential difference \(V\) between the regions of hole and electron conductivity. The hole region is charged positively, the electron region negatively. The potential difference \(V\), caused by the space charge, leads to the passage through the \(p\)-\(n\) junction of a “forward” current \(I_f\).

Fig. 2.

If the external circuit is closed through a resistance \(R\), then a current

\[ I = I_g - I_f \]

flows in it. The equivalent circuit (Fig. 2) consists of a current generator \(I_g\), connected in parallel with the rectifying junction, and a load resistance \(R\). This resistance is the only linear resistance in the simplest case under consideration. The short-circuit current, under the assumptions indicated, is equal to \(I_g\).* The given equivalent circuit, allowing one in the first approximation to analyze the operation of the converter, is sufficient in many cases.

The equations for the currents and voltages in the equivalent circuit may be written as

\[ I_g = I_f + I, \tag{1} \]

\[ I_f = \frac{kT}{qR_0}\left(e^{\frac{qV}{AkT}} - 1\right), \tag{2} \]

\[ IR = V, \tag{3} \]

where \(V\) is the potential difference at the output of the converter. The usual current–voltage characteristic of \(p\)-\(n\) junctions with small ohmic resistance of the semiconductor, which is characterized by rapid “saturation” of \(V\) in the forward direction of current (positive \(V\) and \(I\)), is well represented by equation (2). In particular, it is valid (for \(A = 1\)) for most \(p\)-\(n\) junctions in crystals

\[ \text{*) In the next approximation, in addition to carrier recombination at the surface and in the bulk, it is necessary to take into account the voltage drop and losses in the bulk of the semiconductor and the contact resistances at the electrodes.} \]

germanium. In silicon rectifiers the forward currents, proportional to \(\left(e^{qV/AkT}-1\right)\), increase more slowly \((A>1)\). This form of the characteristic is probably due to a comparatively smooth change in the type of conductivity (a gentle gradient of impurity concentration) and other causes (see, for example, \(^{19}\)).

For a germanium converter the voltage \(V\) is thus related to the current \(I_g\) by the following equation:

\[ \frac{kT}{q}\frac{R}{R_0}\left(e^{\frac{qV}{kT}}-1\right)+V=I_gR. \tag{4} \]

The maximum power is released in the external circuit with a matched load \(R\), which can be determined from the condition

\[ \frac{dW}{dR}=0;\quad W=I^2R=I\cdot V. \tag{5} \]

Hence

\[ \frac{R}{R_0}=e^{\frac{qV}{kT}}. \tag{6} \]

and

\[ I_g=\frac{kT}{qR_0}\ln\left(\frac{R_0}{R}\right)+\frac{kT}{q}\left(\frac{1}{R}-\frac{1}{R_0}\right). \tag{7} \]

Expression (7) makes it possible to determine the matched resistance \(R\) for given \(R_0\) and \(I_g\).

If one introduces the concept of the “reduced current” \(g\), as the ratio of \(I_g\) to the theoretical value of the saturation current in the junction, i.e., set

\[ g=\frac{I_g}{\frac{kT}{qR_0}}=\frac{I_g qR_0}{kT} \tag{8} \]

and denote

\[ \frac{R_0}{R}=Z, \tag{9} \]

then (7) can be written in the form

\[ g=Z\ln Z+Z-1. \tag{10} \]

The dependence of the quantity \(Z\), expressing the relative value of the load conductance, on the reduced current \(g\) is shown in Fig. 3 \(^{12}\); using this dependence, for given \(I_g\), \(R_0\), and \(T\), one can determine the matched load resistance \(R=R_0/Z\), corresponding to the maximum power \(W_m\).

Regardless of the form of the characteristic, the matched resistance \(R\) is equal to the differential resistance of the junction at the given operating point. In the case under consideration it is smaller,

than \(R_j=\dfrac{V}{I_f}\), since

\[ \frac{R}{R_j}=\frac{Z-1}{Z\ln Z}<1. \tag{11} \]

At small \(I_g\), \(R\) tends to \(R_0\), and at \(g=12\) it is \(0.5R_j\) or \(0.2R_0\).

Fig. 3. Dependence of the relative load conductance \(Z\), the reduced maximum power \(\dfrac{W_m}{W_0}\), and the reduced limiting efficiency \(\dfrac{e\eta}{kT}\) on the reduced current \(g\).

Since the current and voltage corresponding to the maximum power are, respectively,

\[ I=\frac{kT}{qR_0}\,Z\ln Z, \tag{12} \]

\[ V=\frac{kT}{q}\,\ln Z, \tag{13} \]

the maximum power is

\[ W_m=\left(\frac{kT}{q}\right)^2\cdot \frac{Z}{R_0}(\ln Z)^2. \tag{14} \]

If we introduce the quantity

\[ W_0=\left(\frac{kT}{q}\right)^2\frac{1}{R_0}, \]

then

\[ W_m=W_0 Z(\ln Z)^2. \tag{15} \]

The dependence of the “reduced” values of the maximum power \(\dfrac{W_m}{W_0}\) on \(g\) is shown in Fig. 3.

The potential difference with the external circuit open is equal to

\[ V_0=\frac{kT}{q}\ln\left(1+\frac{q I_g R_0}{kT}\right). \tag{16} \]

If the ohmic voltage drop in the semiconductor can be neglected, then the dependence of \(V_0\) on \(I_g\) must be the same as the dependence of \(V\) on \(I_f\) in the absence of generation of excess carriers, i.e., it must coincide with the forward branch of the current-voltage characteristic.

The limiting efficiency \(\eta\) can be calculated under the assumption that there is no recombination in the semiconductor, i.e., that all generated excess carriers pass through the \(p\)-\(n\) junction. If an energy \(\varepsilon\) is expended to form one pair of carriers, then, since each carrier passes through the potential difference \(V\), and the fraction of carriers equal to \(I/I_g\) constitutes the load current,

\[ \eta=\frac{qV}{\varepsilon}\cdot\frac{I}{I_g} =\frac{qV^2}{R\varepsilon I_g} =\frac{kT}{\varepsilon}\,Z(\ln Z)^2\,\frac{1}{g}. \tag{17} \]

In contrast to the first two expressions for \(\eta\), the latter is valid provided that a matched resistance \(R\) is included in the circuit. This expression may be rewritten in a form more convenient for analysis:

\[ \frac{\eta\varepsilon}{kT}=\frac{Z(\ln Z)^2}{g}=\frac{W_m}{W_1}, \tag{18} \]

where \(\frac{\eta\varepsilon}{kT}\) is the reduced limiting efficiency, and

\[ W_1=\frac{kT}{q}I_g=gW_0. \tag{19} \]

The dependence of the reduced efficiency on \(g\) is shown in Fig. 3.

The quantity \(W_1\) is convenient to use as a unit of power, since it is proportional to \(I_g\) and makes it possible to determine from the curve of Fig. 3 the maximum power \(W_m\).

The conversion efficiency is small in the case of low radiation energies and increases with \(I_g\).

In the case of fast electrons, according to experimental data,\(^{13}\) \(\varepsilon\) is approximately \(3\) eV for germanium and \(3.6\) eV for silicon.

If these averaged values are used (which, of course, is an approximation and may lead to errors in the case of initial energies of fast electrons that differ greatly from those for which the indicated values were obtained), then \(I_g\) and \(g\) turn out to be simply proportional to the power of the absorbed flux of \(\beta\)-particles.

In the case of conversion of light energy, \(\varepsilon\) is equal to the photon energy \(\frac{hc}{\lambda}\), since the quantum yield in the main absorption band ...

is equal to unity. For any spectral distribution an average energy \(\bar{\varepsilon}\) can be chosen, corresponding to the average wavelength \(\bar{\lambda}\).

The question of the efficiency of conversion of the Sun’s radiation and of the dependence of the efficiency on the absorption coefficient has been considered in detail in works \(^{13,14}\). In calculating the conversion efficiency it is necessary to take into account reflection at the surface of the semiconductor; for germanium, in the wavelength interval from 0.4 to 2 microns, it is about 0.45—0.37. If the radiation enters through the surface into a sufficiently thick layer of semiconductor and is completely absorbed in it (within the principal absorption band, limited by wavelengths of about 2 microns in Ge and 1.2 microns in Si; this already takes place for a layer thickness of the order of 50 microns), the distribution curve must be multiplied by \((1-r)\), where \(r\) is the reflection coefficient.

All the expressions given above did not take into account recombination of excess charge carriers at the surface and in the volume of the semiconductor. Analytical methods of allowing for surface recombination are associated with cumbersome calculations; volume recombination in the case of a one-dimensional problem (normal incidence of light on a plane \(p\)-\(n\) junction) is not difficult to take into account through the diffusion lengths of the minority carriers, as was done in work \(^{13}\). For \(\beta\)-radiation the problem is greatly complicated both by the as yet insufficiently studied processes at the surface and by the fact that absorption of particles is accompanied by strong scattering.

Examples of quantitative estimates

a) Energy of radioactive decay. In the case of absorption by a silicon crystal of \(\beta\)-particles emitted by a thin layer of \(\mathrm{Sr}^{90}—\mathrm{Y}^{90}\), with a source strength of 50 millicuries, one may put \(I_g=10^{-4}\ \text{a}\) and, for an area equal to \(1\ \text{cm}^2\), \(R_0=10^6\ \Omega\). At \(T=300^\circ\) abs.

\[ g=3870, \]

\[ Z=500;\qquad R=2000\ \Omega, \]

\[ I=81\ \mu\text{a},\qquad V=0.16\ \text{v}, \]

\[ W_m=13\ \mu\text{w}. \]

The limiting efficiency is \(\eta=3.8\%\).

b) Conversion of the energy of solar radiation. The value of the solar constant at sea level (allowing for absorption of radiation by the atmosphere under a cloudless sky and a zenith distance of the Sun of \(60^\circ\)) is \(^{12}\) \(0.089\ \text{w}\cdot\text{cm}^{-2}\). If, further, one takes into account the reflection coefficient of silicon, equal to 0.32, then the number of photons with energies sufficient to form excess carriers \((\lambda>1.2\,\mu)\) is \(2.47\cdot10^{17}\ \text{cm}^{-2}\cdot\text{sec}^{-1}\), which corresp-

there exists a limiting current \(I_g = 27 \cdot 10^{-3}\ \mathrm{A\cdot cm^{-2}}\). If \(\bar{\lambda}=0.37\ \mu\) and \(\bar{\varepsilon}=3.3\ \mathrm{eV}\), then for \(R_0=10^6\ \Omega\cdot\mathrm{cm}^2\) and \(T=300^\circ\) abs., \(g=1.04\cdot 10^6\), \(Z=8.4\cdot 10^4\), \(R=11.9\ \Omega\cdot\mathrm{cm}^2\), \(I=24\cdot 10^{-3}\ \mathrm{A\cdot cm^{-2}}\), \(V=0.29\ \mathrm{V}\), \(W_m=7\cdot 10^{-3}\ \mathrm{W\cdot cm^{-2}}\), and \(\eta=7.8\%\).

Experimental data for p–n junctions fabricated by diffusion of boron in a thin surface layer of silicon crystals show that their properties approach the estimates given above\(^ {13}\). The current–voltage characteristics of such junctions corresponded to expression (2), with the value \(A=2—3\).

It should be assumed that in the converters described in\(^ {13}\), the probability of surface recombination was sufficiently small.

5. POSSIBLE SOURCES OF \(\beta\)-RADIATION

The amount of energy of fast electrons absorbed by the semiconductor is determined by the power of the source. If this quantity is divided by the mean energy \(\bar{\varepsilon}\) required to liberate a pair of carriers, then the quotient (under the assumption of ideal “collection” of the carriers by the junction) is equal to the flux of carriers through the junction.

The properties of some radioisotopes are given in the table, which, in addition to nuclei with a pure \(\beta\)-spectrum, includes Co\(^ {60}\) and Cs\(^ {137}\)—Ba\(^ {137}\), which emit \(\gamma\)-rays together with \(\beta\)-particles.

Properties of some radioisotopes

The activities were calculated as follows: if all atoms are radioactive, the mass of 1 curie is equal to $\dfrac{CM\tau}{N}\, g \cdot curie^{-1}$, where $M$ is the molecular weight, $C = 3.7 \cdot 10^{10}$ is the number of disintegrations per second for one curie, and $N = 6.03 \cdot 10^{23}$ is Avogadro’s number. The limiting activity of a unit mass, expressed through $\tau_{1/2} = \tau \ln 2$, is

\[ \frac{1.13 \cdot 10^{13}}{M \tau_{1/2}} \; curie \cdot g^{-1}. \]

In a source deposited on a substrate as a thin layer, self-absorption may be neglected, assuming that half of the emitted $\beta$-particles are absorbed by the semiconductor. As indicated above, the layer may be enclosed between two semiconductor plates. When irradiation is from one side, up to 40% of the particles may be scattered in the backward direction[^16]; however, this effect can be reduced to zero by placing a heavy reflector behind the active layer. In the following quantitative estimates, backscattering will not be taken into account.

Self-absorption sharply reduces the fraction of $\beta$-particles emitted by a thick source. The energy distribution of electrons emitted by a thick layer of $\mathrm{Sr}^{90}$—$\mathrm{Y}^{90}$ was calculated in work[^12] on the basis of the theory for a semi-infinite $\beta$-emitter. The initial energy was taken equal to the mean energy of the $\mathrm{Y}^{90}$ electrons, i.e., $0.89 \, MeV$. Transforming the distribution of the outgoing flux by ranges into a distribution by energies and, for a layer density equal to the density of strontium, gives a mean energy of the outgoing electrons equal to $0.57 \, MeV$, and a number of outgoing electrons amounting to $0.03 n \, cm^{-3}$ per $n$ electrons emitted in $1 \, cm^{3}$.

An activity of $41 \, kcurie \cdot cm^{-3}$ ($16 \, kcurie \cdot g^{-1}$) corresponds to a flux of outgoing electrons of $4.3 \cdot 10^{11} \, cm^{-2} \cdot sec^{-1}$ and an energy flux of $2.5 \cdot 10^{10} \, MeV \cdot cm^{-2} \cdot sec^{-1} = 4 \cdot 10^{-3} \, W$. This energy flux is equivalent to the energy flux from a “thin” source with an activity of $1.5 \, kcurie \cdot cm^{-2}$.

6. CHANGE IN THE PROPERTIES OF A SEMICONDUCTOR UNDER THE ACTION OF FAST ELECTRONS

Bombardment by fast electrons causes the appearance in semiconductor crystals of acceptor centers and recombination centers[^17], which apparently are associated with vacant sites and atoms in the interstices of the crystal lattice. The efficiency of energy conversion gradually decreases upon irradiation of the semiconductor, primarily because of the increased probability of recombination of charge carriers at the surface and in the bulk. In addition, the action of fast particles may be manifested in a decrease of $R_{\nu}$; upon irradiation of the surface with electron conductivity, a counter e.m.f. junction may form near it if, in the layer near the surface, the type of conductivity changes.

In accordance with the theory, it has been shown that at electron energies below a certain threshold \(E_t\), equal for germanium to \(0.63\) MeV \({}^{18}\), long-lived lattice defects are not formed.

According to an estimate not yet confirmed by experiment, for Si the limiting energy \(E_t\) is \(0.3\) MeV \({}^{12}\).

The rate of accumulation of lattice defects in Ge under the action of \(Sr^{90}—Y^{90}\) radiation can be estimated if only the electrons emitted by \(Y^{90}\) are taken into account, since the maximum energy of the electrons from \(Sr^{90}\) is sufficiently small. The energy-distribution spectrum of \(Y^{90}\) electrons is shown in Fig. 4. The mean kinetic energy of the electrons is \(0.89\) MeV. In the case of a thin source, the number of defects produced, referred to the number of electrons, is \(0.01\). The number of defects formed in 1 second by electrons from \(1\) curie of \(Y^{90}\) per \(1\ \text{cm}^2\) will be \(3.6 \cdot 10^8\ \text{cm}^{-2}\cdot\text{sec}^{-1}\), and they are formed to a depth of \(0.156\ \text{cm}\). The average rate of increase of the defect concentration is equal to \(2.3 \cdot 10^9\ \text{cm}^{-3}\cdot\text{sec}^{-1}\). If the effective recombination cross section of electrons and holes is taken to be \(10^{-16}\ \text{cm}^2\) and the thermal velocity equal to \(10^7\ \text{cm}\cdot\text{sec}^{-1}\), the mean lifetime \(\tau\) after \(t\) seconds of irradiation (meaning \(\tau\) due only to the defects that have arisen) will be \([(2.3 \cdot 10^9 t)(10^7)(10^{-16})]^{-1}\) sec.

Fig. 4. Spectrum of \(Y^{90}\) β-particles.

Thus, if the estimate is correct, after 1 day \(\tau\) will be equal to \(5 \cdot 10^{-6}\) sec. An absorber having a thickness sufficient to reduce the limiting energy to a value below \(E_t\), at an activity of \(1\ \text{curie}\cdot\text{cm}^{-2}\), reduces the conversion efficiency to a value on the order of \(10^{-6}\) percent. According to the calculations of the authors of work \({}^{12}\), the use of a thinner absorber, for which the rate of accumulation of defects decreases to a value making the service life of the p-n junction comparable with the half-life of \(Sr^{90}\), leads to a decrease of the efficiency to \(10^{-4}\) percent.

The estimate of the limiting energy \(E_t\) for silicon was made under the assumption that the minimum energy \(E_L\) which must be transferred to an atom in order to move it from a lattice site into an interstice is about \(30\) eV, i.e., equal to the energy calculated for germanium from the experimentally determined value of \(E_t\) according to the relation

\[ \frac{E_L}{mc^2} = 2\frac{m}{M} \left[ \left(\frac{E_t}{mc^2}\right)^2 + 2\left(\frac{E_t}{mc^2}\right) \right], \tag{20} \]

where \(m\) and \(M\) are the masses of the electron and of the atom, \(E_t = 0.63\) MeV. This assumption seems justified, since the structure is the same, the lattice constants are close, and the binding energy\({}^{12}\) is considered to be the same. Apparently, the obtained value of the threshold energy, equal to 0.3 MeV, is correct to an accuracy of about 20%.

7. EXPERIMENTAL DATA

Although the question of using semiconductors for the conversion of radiation energy has been studied only since 1953, by the present time a considerable number of results have been published. Since the latter sometimes contradict one another, below only data that seem sufficiently reliable and that do not have the advertising character inherent in many American publications will be considered.

Fig. 5.

In all published works the source of \(\beta\)-radiation consisted of \(\mathrm{Sr}^{90}—\mathrm{Y}^{90}\) preparations with an activity on the order of tens of millicuries.

In the experiments described in \({}^{12}\), and carried out with both silicon and germanium p–n junctions, the source of particles was a thin layer of carbonaceous salts of \(\mathrm{Sr}^{90}—\mathrm{Y}^{90}\), deposited on the surface of a circle 5 mm in diameter (Fig. 5). The energy flux of the \(\beta\)-radiation was reduced by a protective metal layer to 16% of its initial value.

Silicon and germanium crystals with p–n junctions were tested as converters. The silicon “element” was fabricated by the method of thermal diffusion of lithium, which is a donor impurity\({}^{19}\), into a silicon crystal with hole conductivity. The area of the junction was \(0.15\ \mathrm{cm}^2\), and the depth (the distance from the surface from which the \(\beta\)-particles penetrated) was about \(20\ \mu\). The thickness of the silicon plate was \(0.5\) mm. The diffusion length of the electrons in the crystal before fabrication of the junction was about \(0.02\) cm. About 15% of the energy of the entire flux of \(\beta\)-particles was absorbed within the limits of the diffusion length from the junction.

Figure 6 gives the load characteristics of the converter. The maximum power at \(25^\circ\mathrm{C}\) was \(0.13\cdot 10^{-6}\) W, and at \(-11^\circ\mathrm{C}\) \(0.3\cdot 10^{-6}\) W; if the ohmic losses in the silicon and at the contacts are taken into account, adding the corresponding power to the power released in the load, then at \(25^\circ\mathrm{C}\), \(W_m = 0.23\cdot 10^{-6}\) W. Pos-

It is useful to introduce three differently defined efficiencies: $\eta_1$—the efficiency referred to the total power of the source, $\eta_2$—the efficiency with respect to the energy absorbed by the semiconductor, and $\eta_3$—the efficiency referred to the energy absorbed near the junction (at distances smaller than the diffusion length). In the experiments described, $\eta_1 = 0.04\%$, $\eta_2 = 0.5\%$, $\eta_3 = 2\%$.

The first two quantities point to the unfavorable geometrical conditions of the experiment. It is significant, however, that $\eta_3$ is close to the limiting efficiency, which for the given junction was $2.4\%$.

Fig. 6. Conversion of the energy of Sr $\beta$-particles by a silicon p–n junction.

With decreasing temperature, the power delivered to the load increases for two reasons: $R_0$ increases, owing to the fact that the potential difference grows, and the diffusion length, which determines the fraction of charge carriers effectively used, also increases.

In Rappaport’s experiments$^{13}$, carried out with silicon junctions and a source Sr$^{90}$—Y$^{90}$ of 50 millicuries, the efficiency reached $0.4\%$; the limiting efficiency, according to the author’s estimate, is about $2\%$.

Experiments with germanium crystals with p–n junctions, fabricated both by the method of adding an impurity during crystal pulling and by alloying with indium$^{1}$, showed that the efficiency of energy conversion at the same source intensity, as was to be expected, is considerably lower because of the small values of $R_0$. Figure 7 gives the load characteristics for a grown germanium crystal with $R_0 = 1000$ ohms. The efficiencies, determined in the same way as in the case of silicon, were: $\eta_1 = 0.001\%$, $\eta_2 = 0.009\%$, $\eta_3 = 0.021\%$.

V. S. VAVILOV

Lowering the temperature of the germanium p–n junction from \(25^\circ\mathrm{C}\) to \(-10^\circ\mathrm{C}\) led to an increase in the output power to \(0.8 \cdot 10^{-7}\) W, i.e., by a factor of 24.

In the experiments described, the mean energy of the absorbed \(\beta\)-particles was lower than \(E_t\) for Ge; however, a considerable number of particles had energy above \(E_t\), and accumulation of lattice defects must have occurred. Indeed, at the end of the first two days the short-circuit current \(I_0\) of the silicon “element” had fallen to half

Fig. 7. Conversion of the energy of \(\beta\)-particles of \(S_{2}^{90} — Y^{90}\) by a germanium p–n junction.

of its initial value. However, after 35 days of storage at room temperature (in the absence of a particle flux) and after annealing at \(110^\circ\mathrm{C}\) for 24 hours, the original properties of the junction were completely restored. Together with theoretical considerations on self-restoration of the structure (see \(^{20}\)), these results apparently indicate the possibility of simple methods for restoring the properties of semiconductors irradiated by electrons.

The most reliable data on the efficiency of conversion of solar-radiation energy in large-area silicon junctions appear to be the values reported in works \(^{11}\) and \(^{12}\), where a power of the order of 75 W from \(1\ \mathrm{m}^2\) of the area of a battery is indicated, the battery consisting of many plates, the area of each of which was equal to several square centimeters. If this power is referred to the solar constant at sea level, the efficiency is \(8.3\%\).

CONCLUSION

The semiconductors Ge and Si and the experimental designs using them that have been considered above do not exhaust the possibilities for applying the new method of converting radiation energy. However, already on the basis of the available data it is possible

to judge the directions of development and the field of application of new types of sources of electrical energy.

Let us return to a concrete example of the use of the energy of the β-decay of fission fragments \((\mathrm{Sr}^{90}—\mathrm{Y}^{90})\). At a specific activity equal to 16 curie·gram\(^{-1}\), the power of the flux of β-particles emitted by a thick active layer is \(2.5\cdot 10^{10}\) MeV cm\(^{-2}\) sec\(^{-1}\). If all the excess charge carriers arising in the semiconductor are used, the energy flux corresponds to a current \(I_g=0.0013\) a·cm\(^{-2}\); in germanium \((R_0=10^3\) ohm·cm\(^2)\), whence \(W_m=65\) μW·cm\(^{-2}\) and \(\eta=1.7\%\). In silicon, for \(R_0=10^8\) ohm·cm\(^2\), \(I_g=0.0011\) a·cm\(^{-2}\), \(W_m=340\) μW·cm\(^{-2}\) and \(\eta=8.6\%\). The indicated limiting efficiencies are calculated on the assumption that all the energy of the β-particles is used, which is impossible in the case of the application of a single \(p\)-\(n\) junction. However, by using thin active layers and crystals with \(p\)-\(n\) junctions arranged alternately, efficiencies close to the calculated ones apparently may be achieved.

A decrease in efficiency due to changes in the crystal lattice can be avoided by using isotopes with a lower limiting electron energy, absorbers, or restorative annealing. It was shown above that the absorber method, owing to the form of the β-spectra, is energetically disadvantageous.

For silicon \(p\)-\(n\) junctions the limiting efficiency of conversion of solar radiation is 18% at normal incidence. Since in this case the energy flux is approximately five times greater than the annual average value, which takes into account night, weather, and other factors \(^{12}\), the average electrical power from one square centimeter of converter surface will be about \(3\cdot 10^{-3}\) W.

Already at the present time silicon “solar batteries” can be successfully used for supplying amplifier devices on long-distance telephone lines, portable radio transmitters, for charging storage batteries, and in a number of other cases \(^{15}\).

The conversion efficiency can be increased by increasing

\[ g=\frac{q I_g R_0}{kT}, \]

i.e., by increasing \(R_0\) or \(I_g\), and also by lowering the temperature. A substantial increase in the voltage can be achieved by using \(p\)-\(n\) junctions in which \(V\) increases with direct current more rapidly than in the case described by equation (2), i.e., if the value \(A\) exceeds unity.

CITED LITERATURE

1. Collection Semiconductor Devices, edited by A. V. Rzhanov, I.L. (1953).
2. A. F. Ioffe, “Our Tasks in the Field of Semiconductors,” Izv. AN SSSR 16, 10 (1950); Semiconductors, Publishing House of the Academy of Sciences of the USSR, 1955.
3. H. Moseley, Proc. Phys. Soc. A88, 471 (1913).

1. Lobanov and Belyakov, DAN 47, No. 5 (1945).
2. P. Rappaport, E. Linder, J. Appl. Phys. 24, 1110 (1953).
3. J. Kramer, The Electrician 93, 497 (1924).
4. P. Ohmart, J. Appl. Phys. 22, 1504 (1951).
5. W. A. Ulianin, Wied. Ann. 34, 241 (1888).
6. Chapin, Fuller, Pearson, J. Appl. Phys. 25, 676 (1954).
7. Collection Semiconductor Materials, edited by V. M. Tuchkevich, IL (1954).
8. In the collection Practical Use of Radioisotopes, published by the Stanford Research Institute in 1951, the cost of 1 curie of Sr$^{90}$—Y$^{90}$ at an activity of 33 curies·g, is given as equal to 2 dollars.
9. W. Pfann, W. Van Roosbroeck, J. Appl. Phys. 25, 1422 (1954).
10. P. Rappaport, Phys. Rev. 93, 426 (1954).
11. R. Cummerow, Phys. Rev. 95, 16 (1954); 95, 561 (1954).
12. Bell Laboratories Record 32, No. 6, 233 (1954).
13. H. Seliger, Phys. Rev. 88, 408 (1952).
14. See$^{1}$, the article by Shockley and Read, “Statistics of the Recombination of Nonequilibrium Charge Carriers.”
15. E. Klontz, K. Lark-Horovitz, Phys. Rev. 82, 763 (1951). Also see the article in 10 86, 643 (1952).
16. C. Fuller, J. Ditzenberger, Phys. Rev. 91, 193 (1953).
17. Brown, Fletcher, Wright, Phys. Rev. 92, 591 (1953).
18. Zh. I. Alferov, B. M. Konovalenko, S. M. Ryvkin, V. M. Tuchkevich, A. I. Uvarov, ZhTF 25, 11 (1955).
19. B. M. Konovalenko, S. M. Ryvkin, V. M. Tuchkevich, ZhTF 25, 18 (1955).
20. S. M. Ryvkin, ZhTF 25, 21 (1955).
21. A. F. Ioffe, Semiconductors in Modern Physics, Publishing House of the Academy of Sciences of the USSR, 1955.