IRRADIATION OF SEMICONDUCTORS WITH NUCLEONS

K. Lark-Horovitz*)

In simple semiconductors, such as, for example, germanium and silicon, a definite number of impurity centers can be deliberately introduced; in doing so, on the basis of the type and number of introduced impurities, one can predict both the number of carriers (per impurity atom) and their sign. Thus, at will, one can obtain semiconductors of the n-type (electronic) or p-type (hole), possessing a conductivity whose value may vary over wide limits.

Semiconductors with specified electrical properties can also be obtained by heating. If germanium is heated to approximately 850° C and then rapidly cooled, p-type germanium is formed. If, however, this same substance is heated for several hours at a temperature of 450° C, n-type germanium is obtained. Heating and rapid cooling of silicon at first creates a high resistance. Finally, temporary changes in conductivity can be produced as a result of introducing carriers either by bombardment with electrons or by introducing holes through the boundary of the semiconductor at its contact with a metal. These changes disappear after definite intervals of time—of the order of microseconds—after removal of the field. Under heating or under the action of mechanical stresses the effect is reversible; the introduction of impurity centers, however, causes an irreversible change in the electrical properties. By introducing impurity centers of the opposite type, one can obtain such an equilibrium state in which holes and electrons “neutralize” one another and the substance consequently acquires a very high resistance. These centers can be removed and the properties of the pure material restored only by means of special chemical treatment.

All these effects may be caused by the passage of nucleons (deuterons, protons, neutrons, or α-particles), and some of them by the passage of electrons through a semiconductor. The creation of artificial radioactivity leads to the appearance of final decay products acting as impurity centers. Thus—

*) Semi-Conducting Materials, London, 1951, p. 47.

thus, if the type and cross section of the reaction are known, new types of semiconductors with the desired properties can be obtained.

Elastic collisions between the traversing particles and the atoms of the semiconductor cause the appearance of defects, vacancies, and atoms in interstices of the lattice, thus forming new energy levels and freeing carriers of various signs. These lattice defects also act as scattering centers. It is possible that a “point heating” takes place here, i.e. that small regions of the substance melt and then are suddenly cooled. Here one observes not only phenomena close to those caused by heat treatment, but it is also possible that surface states are created and “frozen in”[^13] at the boundary between the solid and the molten substance. Finally, the passage of electrons of medium energy or of γ-radiation through a substance possessing high resistivity causes temporary effects[^7]. If, however, the energy is sufficiently high, then even in the case of electrons the momentum transferred to the bombarded atoms may prove sufficiently large to cause disordering of the lattice[^14]. The study of the behavior of semiconductors under the action of nucleons is especially interesting because the number of carriers present in a semiconductor is so small that even brief irradiation by a flux of moderate intensity can cause effects readily observable from changes in the electrical properties of the substance.

THE EFFECT OF NUCLEAR TRANSMUTATIONS ON SEMICONDUCTORS

First of all we shall consider new semiconductors of constant type, obtained by means of nuclear transmutations in a substance.1

For such elemental semiconductors as, for example, boron, silicon, germanium, selenium, and tellurium, the total cross section for activation by thermal neutrons is known, and it is therefore easy to judge whether irradiation can cause appreciable effects. The conduction mechanism of boron is still not fully understood. Tellurium and selenium have been studied in great detail; it remains, however, not entirely clear how impurities affect their conductivity. We have not yet succeeded in obtaining such tellurium samples as would possess $n$-type conductivity at all temperatures.

Silicon and germanium are well-known substitutional semiconductors; however, silicon has a total absorption cross section for thermal neutrons of only $\sigma_A = 0.1 \cdot 10^{-24}\ \mathrm{cm}$, and the only $(n,\gamma)$ process leading to the formation of impurity centers is

$$ \mathrm{Si}^{30}(n,\gamma)\mathrm{Si}^{31} \to \mathrm{P}^{31} $$

with $\mathrm{Si}^{30}$ present in an amount of only 3.08%. Fast neutrons can lead to the formation of $\mathrm{Al}^{27}$ as a result of the reaction

$$ \mathrm{Si}^{28}(n,2n)\mathrm{Si}^{27} \to \mathrm{Al}^{27}. $$

A list of possible nuclear transmutations in germanium is given in Table I.

Table I

Summary of nuclear reactions in germanium

Content of isotope, %	Isotope
		I. Principal reactions with deuterons and slow neutrons	I. Principal reactions with deuterons and slow neutrons
		\((d,p)\), \((n,\gamma)\)	\((d,n)\)
21.2	\(\mathrm{Ge}^{70}\)	\(\mathrm{Ge}^{71}\xrightarrow{11.4\ \text{days}}\mathrm{Ga}^{71}\)	\(\mathrm{As}^{71}\xrightarrow{50\ \text{h}}\mathrm{Ge}^{71}\xrightarrow{11\ \text{days}}\mathrm{Ga}^{71}\)
27.3	\(\mathrm{Ge}^{72}\)	\(\mathrm{Ge}^{73}\)	\(\mathrm{As}^{73}\xrightarrow{76\ \text{days}}\mathrm{Ge}^{73}\)
7.9	\(\mathrm{Ge}^{73}\)	\(\mathrm{Ge}^{74}\)	\(\mathrm{As}^{74}\xrightarrow{17.5\ \text{days}}\begin{matrix}\mathrm{Ge}^{74}\\ \mathrm{Se}^{74}\end{matrix}\)
37.1	\(\mathrm{Ge}^{74}\)	\(\mathrm{Ge}^{75}\xrightarrow{82\ \text{min}}\mathrm{As}^{75}\)	\(\mathrm{As}^{75}\)
6.5	\(\mathrm{Ge}^{76}\)	\(\mathrm{Ge}^{77}\xrightarrow{12\ \text{h}\ 59\ \text{s}}\mathrm{As}^{77}\xrightarrow{40\ \text{h}}\mathrm{Se}^{77}\)	\(\mathrm{As}^{77}\xrightarrow{40\ \text{h}}\mathrm{Se}^{77}\)
		II. Principal reactions with fast neutrons	II. Principal reactions with fast neutrons
		\((n,p)\)	\((n,2n)\)
21.2	\(\mathrm{Ge}^{70}\)	\(\mathrm{Ga}^{70}\xrightarrow{20\ \text{min}}\mathrm{Ge}^{70}\)	\(\mathrm{Ge}^{69}\xrightarrow{40\ \text{h}}\mathrm{Ga}^{69}\)
27.3	\(\mathrm{Ge}^{72}\)	\(\mathrm{Ga}^{72}\xrightarrow{14.3\ \text{h}}\mathrm{Ge}^{72}\)	\(\mathrm{Ge}^{71}\xrightarrow{11.4\ \text{days}}\mathrm{Ga}^{71}\)
7.9	\(\mathrm{Ge}^{73}\)	\(\mathrm{Ga}^{73}\xrightarrow{5\ \text{h}}\mathrm{Ge}^{73}\)	\(\mathrm{Ge}^{72}\)
37.1	\(\mathrm{Ge}^{74}\)	\(\mathrm{Ga}^{74}\xrightarrow{?}\mathrm{Ge}^{74}\)	\(\mathrm{Ge}^{73}\)
6.5	\(\mathrm{Ge}^{76}\)	\(\mathrm{Ga}^{76}\xrightarrow{?}\mathrm{Ge}^{76}\)	\(\mathrm{Ge}^{75}\xrightarrow{82\ \text{min}}\mathrm{As}^{75}\)

Content of isotope, %	Isotope	\((n,\alpha)\)
21.2	\(\mathrm{Ge}^{70}\)	\(\mathrm{Zn}^{67}\)
27.3	\(\mathrm{Ge}^{72}\)	\(\mathrm{Zn}^{69}\xrightarrow{14\ \text{h}\ 52\ \text{m}}\mathrm{Ga}^{69}\)
7.9	\(\mathrm{Ge}^{73}\)	\(\mathrm{Zn}^{70}\)
37.1	\(\mathrm{Ge}^{74}\)	\(\mathrm{Zn}^{71}\xrightarrow{2.2\ \text{m}}\mathrm{Ga}^{71}\)
6.5	\(\mathrm{Ge}^{76}\)	\(\mathrm{Zn}^{73}\xrightarrow{<2\ \text{m}}\mathrm{Ga}^{73}\xrightarrow{5\ \text{h}}\mathrm{Ge}^{73}\)

Continuation of Table I

Isotope content, %	Isotope	(α, n)	(α, 2n)	(α, p)
		III. Principal reactions with α-particles	III. Principal reactions with α-particles	III. Principal reactions with α-particles
21.2	Ge^70	Se^73 \(\xrightarrow{71\ \mathrm{h}}\) As^73 \(\xrightarrow{76\ \mathrm{d}}\) Ge^73	Se^72 \(\xrightarrow{9.7\ \mathrm{d}}\) As^72 \(\xrightarrow{26\ \mathrm{h}}\) Ge^72	As^73 \(\xrightarrow{76\ \mathrm{d}}\) Ge^73
27.3	Ge^72	Se^75 \(\xrightarrow{127\ \mathrm{d}}\) As^75	Se^74	As^75
7.9	Ge^73	Se^76	Se^75 \(\xrightarrow{127\ \mathrm{d}}\) As^75	As^76 \(\xrightarrow{26.8\ \mathrm{h}}\) \(\begin{cases}\mathrm{Ge}^{76}\\ \mathrm{Se}^{76}\end{cases}\)
37.1	Ge^74	Se^77	Se^76	As^77 \(\xrightarrow{40\ \mathrm{h}}\) Se^77
6.5	Ge^76	Se^79 \(\xrightarrow{?}\) Br^79	Se^78	As^79 \(\xrightarrow{?}\) Se^79 \(\xrightarrow{?}\) Br^79

In the case of germanium the total cross section for activation by thermal neutrons is \(2.3 \cdot 10^{-24}\ \mathrm{cm}^2\). Pomerantz^15 recently determined the isotopic cross section for oxides of separated germanium isotopes and obtained the values given in Table II. On the basis of these values,

Table II

Absolute capture cross section of separated germanium isotopes

Isotope	Amount, %	Capture cross section in \(10^{-24}\ \mathrm{cm}^2\), isotopic	Capture cross section in \(10^{-24}\ \mathrm{cm}^2\), atomic	Final product
Ge^70	21.2	3.25	0.69	Ga
Ge^72	27.3	0.94	0.26	Ge
Ge^73	7.9	13.69	1.08	Ge
Ge^74	37.1	0.60	0.22	As
Ge^76	6.5	0.35	0.02	Se

one may suppose the formation of \(p\)-type germanium as a result of nuclear transmutations, since the amount of gallium formed is pre-

increases the amount of arsenic formed. From the measured cross sections one can calculate the excess of the number of acceptors (gallium atoms) over the number of donors (arsenic atoms), using the formula

\[ N_e = n_n vt \cdot \overline{\sigma_e}\cdot N_{\mathrm{Ge}}\cdot P_i, \tag{1} \]

where \(N_e\) is the number of observed events, \(n_n vt\) is the integral flux of neutrons, \(\overline{\sigma_e}\) is the cross section of the process, \(N_{\mathrm{Ge}}\) is the number of germanium atoms in \(cm^3\), and \(P_i\) is the percentage of atoms of isotopes of the given kind. Consequently, if a germanium sample is subjected to the action of the full flux \(n_n vt\), then the excess of acceptors created in \(1\ cm^3\) of the sample will be approximately equal to \(2\%\) of this number. If each impurity atom releases only one carrier (as we assume in our studies of electrical properties\(^{1-3}\)), then this can be checked

Fig. 1a. Hall coefficient as a function of reciprocal temperature of \(n\)-type germanium bombarded with fast and slow neutrons. (Note the effect of heat treatment: it causes restoration of \(n\)-type germanium.)

\(I\) — after exposure (\(p\)-type),
\(II\) — after reaching \(400^\circ C\),
\(III\) — after two additional hours at \(400^\circ C\),
\(IV\) — after heating to \(450^\circ C\).

in the following way2. Germanium samples with a known quantity of impurity centers (of the \(p\)- or \(n\)-type) are placed for a certain time \(t\) in a nuclear reactor in the path of a known flux of thermal neutrons. After exposure the samples are heated to \(450^\circ\mathrm{C}\) for 24 hours (or until further heat treatment ceases to produce any new changes). Then

Fig. 16. Resistance of a germanium sample irradiated with neutrons, as a function of inverse temperature. (Note the strong influence of impurity scattering before heat treatment and the seemingly high value of the activation energy.)

\(I\) — after irradiation with neutrons (\(p\)-type),
\(II\) — after heating to \(400^\circ\mathrm{C}\),
\(III\) — after two additional hours at \(400^\circ\mathrm{C}\),
\(IV\) — after heating to \(450^\circ\mathrm{C}\).

the sample is slowly cooled to room temperature in order to avoid the formation of lattice defects possible under rapid cooling, and to eliminate all disorder phenomena in the lattice due to fast neutrons. If the purest germanium is used in the form of single crystals, then, provided that all lattice-order disturbances caused by fast neutrons have been eliminated, we obtain an “ideal” semiconductor, since the newly introduced atoms are precisely localized at lattice sites. Sometimes collisions may change their position and cause additional disorder, which, however, disappears after heat treatment. As a result, the value of the conductivity is determined by the balance between carriers released from atoms of the initially present impurities of one sign or another and impurity atoms that have appeared as a result of transmutations. The accompanying figures illustrate the character of the Hall effect (Fig. 1a) and the behavior of the specific resistance (Fig. 16) after bombardment

doping and heat treatment of the sample. The final curves for the Hall effect are typical of impurity semiconductors obtained by substitution, whereas the curves obtained immediately after exposure (under irradiation with slow and fast neutrons) and in samples with the disorder in the lattice produced as a result of quenching indicate a substantially more complex behavior of the substance.

Table III gives a summary of the results and shows that, within the experimental errors (the cross section and the flux are known only with an accuracy of up to 20%), the number of liberated carriers determined from measurements of the Hall effect is in fact equal to the number of impurity centers calculated from the activation cross section and the flux data.

Table III

Concentration of impurity atoms appearing in germanium as a result of transmutations

Sample number	Initial number of carriers in \(1\ \mathrm{cm}^3\), \(n_0\)	Total flux \((n_nvt)\)	Final number of carriers in \(1\ \mathrm{cm}^3\), calculated from the cross section	Final number of carriers in \(1\ \mathrm{cm}^3\), measured by the Hall effect	Difference, %
1 (\(n\)-type)	\(-5.25\cdot 10^{14}\)	\(+4.56\cdot 10^{17}\)	\(+8.82\cdot 10^{15}\)	\(+6.16\cdot 10^{15}\)	\(-27\)
2 (\(n\)-type)	\(-2.28\cdot 10^{16}\)	\(+2.54\cdot 10^{18}\)	\(+2.92\cdot 10^{16}\)	\(+3.21\cdot 10^{16}\)	\(+9\)
3 (\(n\)-type)	\(-5.47\cdot 10^{16}\)	\(+4.37\cdot 10^{18}\)	\(+3.48\cdot 10^{16}\)	\(+5.25\cdot 10^{16}\)	\(+33\)
4 (\(n\)-type)	\(-1.17\cdot 10^{16}\)	\(+2.54\cdot 10^{18}\)	\(+4.03\cdot 10^{16}\)	\(+5.20\cdot 10^{16}\)	\(+22\)
5 (\(n\)-type)	\(-1.48\cdot 10^{14}\)	\(+2.54\cdot 10^{18}\)	\(+5.05\cdot 10^{16}\)	\(+4.45\cdot 10^{16}\)	\(-13\)
6 (\(p\)-type)	\(+5.02\cdot 10^{14}\)	\(+2.54\cdot 10^{18}\)	\(+5.15\cdot 10^{16}\)	\(+4.95\cdot 10^{16}\)	\(-4\)
7 (\(p\)-type)	\(+4.60\cdot 10^{14}\)	\(+8.04\cdot 10^{17}\)	\(+6.25\cdot 10^{16}\)	\(+5.22\cdot 10^{16}\)	\(-20\)
8 (\(n\)-type)	\(-4.04\cdot 10^{14}\)	\(+1.06\cdot 10^{19}\)	\(+2.15\cdot 10^{17}\)	\(+1.90\cdot 10^{16}\)	\(-13\)

Since semiconductors obtained as a result of nuclear transmutations possess quite definite properties which cannot be changed by physical treatment, it is possible to create a semiconductor with high resistance by irradiating an initial substance of \(n\)-type at a certain increase of \(n_nvt\), and thereby to introduce a measurable amount of excess \(p\)-type material for “neutralization.” Such a semiconductor will have a somewhat lower conductivity than an intrinsic semiconductor (owing to the presence of additional scattering centers and, accordingly, a smaller mean free path), but the same slope of the curve characteriz-

Fig. 2. Resistance as a function of reciprocal temperature for germanium irradiated with neutrons upon gradual increase in the number of nuclear transmutations. (The resistance keeps increasing, whereas the Hall coefficient decreases; cf. Fig. 3.)

I — before exposure (n-type),
II — after 4.35-hour exposure followed by heating,
III — ” 5.85 ” ” ” ” ”
IV — ” 6.85 ” ” ” ” ”
V — ” 8.35 ” ” ” ” ”
VI — ” 10.85 ” ” ” ” ”

Fig. 3. Hall coefficient as a function of reciprocal temperature for “neutralized” n-type germanium, upon the subsequent nuclear transmutations. The achieved maximum value coincides with the theoretical one (cf. Appendix 1 and Fig. 10).

I — before exposure (n-type),
II — after 4.35-hour exposure followed by heating,
III — ” 5.85 ” ” ” ” ”
IV — ” 6.85 ” ” ” ” ”
V — ” 8.35 ” ” ” ” ”
VI — ” 10.85 ” ” ” ” ”

...dependence of the logarithm of the resistance on \(1/T\) (Fig. 2). Fig. 3 illustrates the corresponding change in the Hall coefficient.

These newly introduced donors and acceptors are equally effective. The ratio of the tangents of the angles of inclination of the conductivity curves—irradiation should have been equal to the ratio of the mobilities \((\sigma_e=en_eb_e,\ \sigma_h=en_hb_h,\) where \(e\) is the charge, \(n_e\) and \(n_h\) are the numbers of electrons and holes, and \(b_e\) and \(b_h\) are their mobilities). Since the newly introduced impurity atoms remain in the crystal, the effect of “healing” of the lattice does not occur here.

Under irradiation with fast neutrons, impurities with smaller atomic numbers usually appear. For example, the reaction \((n,p)\) leads to the appearance of gallium, which then transforms into germanium; the reaction \((n,\alpha)\) gives zinc, which transforms partly into gallium and germanium. The reaction \((n,2n)\) gives germanium, which transforms partly into gallium; and only germanium-76, appearing in insignificant amount, gives arsenic. Therefore, if the slow neutrons are filtered out by introducing cadmium, it would be possible to obtain a more sharply expressed shift in the direction of impurities of the gallium type.

IRRADIATION WITH DEUTERONS

Since the influence of irradiation by nucleons on the conductivity of germanium was first observed after bombardment of germanium with deuterons, it was essential to establish to what extent this effect could be due to the formation of impurities as a result of nuclear transmutations. The number of decay events \(N_d\) caused by one deuteron (with an energy of \(10\) MeV) may be estimated\(^{19}\) approximately as

\[ N_d=\bar{\sigma}_A r_{\mathrm{eff}} N_{\mathrm{Ge}}, \tag{2} \]

where \(\bar{\sigma}_A\) is the cross section for activation, of the order of \(5\cdot10^{-25}\ \text{cm}^2\). In this formula \(r_{\mathrm{eff}}\) is an effective radius, taking into account the decrease of the cross section at lower energies (of the order of \(0.2\ \text{mm}\)), which is approximately equivalent to \(0.1\ \text{g}/\text{cm}^2\); \(N_{\mathrm{Ge}}\) is the number of germanium atoms per gram. Under these assumptions we find that the number of decay events is of the order of \(5\cdot10^{-4}\) per deuteron, i.e., several orders of magnitude smaller than the actual number of carriers, determined from the measurement of the conductivity caused by bombardment with deuterons, and equal to \(1\text{–}2\).

Bleiler and Tendam\(^{19}\) measured the activities caused by irradiation of a germanium sample with deuterons and, following the decay for approximately three weeks, the authors found the presence of six periods. By introducing the appropriate correction factor (taking into account the absorption of \(\beta\)-radiation in the sample), it proved possible to estimate the cross sections, assuming that: 1) the ratio \(K\)-capture/\(\beta^+\) obeys the theo-

Fermi, 2) all isomers appear with equal probability, and 3) the $(d,n)$- and $(d,p)$-cross sections (the Oppenheimer–Phillips process) are identical.

Under these assumptions it was possible to estimate the cross sections for various processes; according to these observations it turned out that one deuteron causes $8.3\cdot 10^{-5}$ reactions
$\mathrm{Ge}^{74}(d,p)\mathrm{Ge}^{75}$. With an isotopic abundance of $37.1\%$ this corresponds to a cross section of $2.3\cdot 10^{-25}\ \mathrm{cm}^2$. Taking, for the process $\mathrm{Ge}^{74}(d,n)\mathrm{As}^{75}$, which does not give rise to activity, a relative cross section equal to 2.5, we obtain the following values for the total number of new nuclei created by one deuteron: a) immediately after bombardment the only impurity is arsenic, formed in the $(d,n)$ process, and the number of arsenic nuclei is $5\cdot 10^{-4}$ per deuteron; b) after the decay of all radioactive nuclei, impurities of gallium, arsenic, and selenium remain in the following amounts:
$N_{\mathrm{Ga}}=2\cdot 10^{-4}$, $N_{\mathrm{As}}=2.9\cdot 10^{-4}$, and $N_{\mathrm{Se}}=0.27\cdot 10^{-4}$. The total number of newly created impurity centers is thus $5.2\cdot 10^{-4}$ per deuteron. A recalculation of the results of these experiments, with the inclusion of new data on induced activities, indicates a somewhat larger amount of selenium formed. The amount of arsenic is again greater than the amount of gallium. This could have been expected even without a detailed calculation, since gallium is formed only as a result of the $(d,p)$ reaction on $\mathrm{Ge}^{70}$, whereas arsenic is the final product of both $(d,p)$ and $(d,n)$ reactions on the more abundant $\mathrm{Ge}^{74}$.

Although these figures are not entirely reliable, owing to the absence of detailed information on the decay schemes, they are in good agreement with the theoretical estimates given above. Thus it is clear that, in order to induce effective nuclear transmutations, sufficiently intense deuteron fluxes are needed; but if such fluxes are available, one may expect that an $n$-type substance will be obtained owing to the excess of donors in the form of arsenic atoms, and not a $p$-type substance, as in neutron irradiation. It is also possible to place the substances under study in a cyclotron, because even if they melt, the number of nuclear transmutations will remain unchanged.

As is seen from the analysis, in the reactions $(\alpha,n)$ and $(\alpha,p)$ only arsenic and selenium are formed. When selenium was added to germanium by chemical means, we did not succeed in obtaining indications of any change in the number of carriers in germanium; from this one may conclude that irradiation with $\alpha$-particles will lead to an $n$-type substance as a result of the formation of arsenic. Thus it may be assumed that, whereas neutron irradiation gives a $p$-type substance, irradiation with charged particles—for example, deuterons and $\alpha$-particles—can lead to the formation of an $n$-type substance.

LATTICE DEFECTS CAUSED BY FAST NEUTRONS AND CHARGED PARTICLES*)

As was already mentioned in discussing the question of activation by deuterons, the effects observed after irradiation with deuterons are so large that they cannot be explained by nuclear transmutations alone. It must therefore be assumed that some other phenomenon is involved here, in connection with which it is necessary briefly to consider the various processes observed when particles pass through the bombarded substance.

In doing so, two questions must be considered. The first is the question of the interaction of the particle with the bombarded substance \(^{20}\), which is a problem of radiation physics \(^{21, 22}\) and is of interest to us only insofar as it is necessary for understanding the properties of the solid state. The most important question here, however, is the nature of the new energy states, if their appearance is caused by irradiation with charged and uncharged particles.

The effects of nuclear irradiation differ depending on the type of particles used. When \(\gamma\)-radiation is used, primary transient effects are observed, such as, for example, the photoelectric effect, the Compton effect, and pair formation. When irradiation is carried out with singly charged particles, the principal processes are ionization and excitation. The stopping power of a substance, or the energy losses for this process, are well known and may be expressed by the formula

\[ -\left(\frac{dW_p}{dx}\right)_{\text{ion}} = 2\pi ZN_s \frac{r_0^2 M_p (mc^2)^2}{mW_p} \cdot \ln \frac{2mv^2}{W_1}, \tag{3} \]

Here \(N_s\) is the number of stationary atoms/\(\mathrm{cm}^3\), \(Z\) is the number of electrons that can be liberated during ionization, \(m\) is the electron mass, \(M_p\) is the mass, \(W_p\) the energy, and \(v\) the velocity of the penetrating particles. \(W_1\) is the mean ionization energy of the atom, and \(r_0\) is the electron radius. As a result of nuclear collisions, the charged particle also loses an energy equal to

\[ -\left(\frac{dW_p}{dx}\right)_{\text{coll}} = \pi N_s Z^2 r_0^2 \frac{M_p}{M_A} \frac{(mc^2)^2}{W_p} \ln \left[ \frac{\Delta W_p(\text{max})}{\Delta W_p(\text{min})} \right]. \tag{4} \]

In this formula \(Z\) is the charge of the stationary nucleus, and \(M_A\) its mass. The maximum energy loss \(\Delta W_p(\text{max})\) is determined by the maximum transfer of energy in any collision, while the minimum energy loss \(\Delta W_p(\text{min})\) is determined by the screening limit. In the case of neu-

*) The experiments were carried out jointly with R. E. Davis, I. Forster, I. Cleland, I. Crawford, V. Johnson, and I. Pigg.

trons (mass \(=M_n\)) the energy loss can be determined in the following way:

\[ \Delta W_p = W_{p0}\frac{4M_nM_A}{(M_A+M_n)^2}\sin^2\frac{\theta}{2} = W_{p0}\frac{2A_W}{(A_W+1)^2}(1-\cos\theta), \tag{5} \]

where \(\theta\) is the angle of deflection in the “center-of-mass system”; in place of \(M_A\) and \(M_n\) the atomic weight \(A_W\) and 1 have been substituted. The average energy loss per collision will be

\[ \overline{\Delta W_p}=W_{p0}\frac{2A_W}{(1+A_W)^2}. \tag{6} \]

It is clear from this that neutrons and heavy charged particles, upon collision, can create lattice defects—vacant sites and atoms in interstitial positions—which substantially alter the band structure of the substance.

The cross section for elastic scattering of deuterons with energy \(W_p\) can be written in the form \(\sigma(W_p,W_r^0)\) and an expression obtained for the scattering cross section at a given minimum recoil energy \(W_r^0\):

\[ \bar{\sigma}(W_p,W_r^0)=\pi Z^2r_0^2\frac{M_D}{M_A}\cdot \frac{(mc^2)^2}{W_pW_r^0} \left(1-\frac{M_AW_r^0}{4M_DW_p}\right), \tag{7} \]

Here \(M_D\) is the mass of the deuteron, and \(W_r^0\) is the minimum recoil energy necessary for displacing the bombarded atom. If the quantity

\[ \frac{M_AW_r^0}{4M_DW_p} \]

is neglected in comparison with unity, then for germanium we obtain:

\[ \bar{\sigma}(W_r^0)=\frac{1.83\cdot10^{-18}}{W_p\cdot W_r^0}\ \text{cm}^2, \tag{8} \]

where \(W_p\) is expressed in MeV, and \(W_r^0\) in eV.

The relation between the energy and the effective radius for deuterons with energy in the range between 2 and 10 MeV for germanium was experimentally determined by Bleyler, Heller, and Tendam and can be written as \(W_p=0.480\cdot r_{\mathrm{eff}}^{0.614}\), where \(W_p\) is given in MeV, \(r_{\mathrm{eff}}\) in \(g/cm^2\). Below 2 MeV the radius is somewhat larger than follows from this formula. Using the relation between the energy and the effective radius and the value of the cross section (8), one can calculate the total number of primary collisions in the specimen. If it is assumed that the minimum recoil energy necessary for displacing an atom is equal to 30 eV, we obtain for the cross section \(0.61\cdot10^{-19}\cdot\dfrac{1}{W_p}\). Using

this value, when considering the relation between the energy and the effective radius, we come to the conclusion that, for deuterons with an energy of \(10\) Mev, the total number of primary collisions is approximately \(25\). Each primary collision may entail secondary collisions, while energy is lost until the transferred momentum becomes too small to produce a displacement of the atom. If one considers a germanium atom as a light fission product, then, applying analogous relations at lower energy, one can calculate that the average number of secondary collisions will be of the order of \(8\). This will lead to a total number of collisions of the order of \(200\). On the other hand, Seitz\(^{21,22}\) has recently extended his earlier calculations of energy losses in collisions to the case of germanium and silicon. Using his method, we find that the total number of collisions caused by deuterons of energy \(10\) Mev is \(175\); this value is in good agreement with the results of our estimate.

The actual number of carriers created by one deuteron is considerably smaller—only \(1\) or \(2\), which indicates the large role of lattice “healing” effects during the bombardment process and during the interval of time that has elapsed before the Hall-effect measurements begin. Even in very early work carried out with \(\alpha\)-particles, we found a considerable “fall-off” of conductivity, and this must be taken into account when considering the results of experiments with deuterons.

Calculations taking into account the presence of a cascade process for germanium atoms displaced by neutrons at a certain definite energy distribution near \(2\) Mev were carried out by Ivens\(^{23}\). Assuming that the lattice distortions are caused exclusively by displacements of germanium atoms produced by elastic collisions with fast neutrons, and also that no other irradiation exerts any influence in this case, and neglecting recombination, back diffusion, and the role of annealing, Ivens treats scattering in a semiconductor as scattering by individual atoms. For each given value of the neutron energy \(W_p\), he assumes the distribution of collision energies to be uniform in the interval from \(0\) to

\[ \frac{4A W}{(1 + A W)^2}, \]

where \(A W\) is the atomic weight of germanium. For each possible value of the neutron energy the number and energy of secondary collisions are determined. In calculating the number of secondary collisions, only those elastic collisions are considered in which an energy of at least \(25\) ev is transferred to the atom, on the assumption that \(25\) ev is the average energy required to displace a germanium atom from its position in a lattice site. (It will be seen below that this assumption is not far from the truth.) As a result, it proves possible to trace the scattering of the energies of the bombarded atoms and to estimate the total number of displaced

germanium atoms. The total number of scattered atoms turns out to be equal to \(1.19 \cdot 10^{-4} W_p\). For one particular case of the energy distribution, about 135 scattered atoms were obtained per collision. Taking the scattering cross section to be \(1 \cdot 10^{-24}\ \text{cm}^2\), we find that the actual number of scattered atoms per incident neutron will be about 6 in \(1\ \text{cm}^3\). This figure is in good agreement with the experimental data obtained later. It should be noted, however, that this seemingly good agreement is to a considerable extent destroyed by a different choice of the cross section. Of course, the fact that other radiation also acts simultaneously, and also that annealing of the lattice takes place, must be taken into account and may appreciably change the results.

On the other hand, the possibility is not excluded that the process of “point heating,” whose existence Dessauer\({}^{24}\) proposed in order to explain the interaction of radiation of high intensity with living matter, may play an essential role. Considering this process in a quite elementary way, one may say that the available energy is sufficient to melt only a small part of the germanium specimen. Upon rapid cooling the latter is transformed from \(n\)-germanium into \(p\)-germanium. Using the values of the thermal conductivity and the latent heat of fusion, one can estimate the amount of substance melted as a result of the collision of atoms possessing the given energy\({}^{25}\). A germanium atom which, after collision with a neutron, has an energy of \(10^5\ \text{eV}\), expends its energy on melting a region of radius approximately \(10^{-6}\ \text{cm}\). However, this region will cool in an extremely short time, \(10^{-12}\ \text{sec.}\)\({}^{26}\), and thus very rapid cooling will take place. The solution of this problem of radiation physics, i.e. the comparison of the action of isolated defects with the action exerted by lattice disturbances caused by point heating, would require a large number of very precise and laborious experiments; discussion of this question lies beyond the scope of the present article. In both cases, irrespective of whether the disorder is caused by inelastic and elastic collisions or by point heating, a certain number of vacant sites and interstitial atoms appear in the lattice. This means that the original lattice is distorted and, consequently, the character of the band structure of the substance has changed.

If it is assumed that vacant sites and interstitial atoms in the germanium and silicon lattice have the same effect as the replacement of germanium atoms by atoms of the third and fifth columns of the periodic system would have, then one may conclude that vacant sites create unoccupied acceptor states situated above the filled band, while interstitial atoms create unoccupied donor states lying below the conduction band. If these states were formed in the same way as in chemical substitution and in equal numbers, then their effect on the conductivity could not be

...found, since they would have had to neutralize one another. The activation energy of impurities due to chemical substitution is so small that practically all donors and all acceptors are ionized at room temperature. However, in the case of displacements caused in the lattice by heat treatment or by collisions, we must assume that there is some distribution of possible values of the activation energies, and if the donor level lies below the acceptor level, then the donors will not be effective. Therefore the type of carrier obtained depends on the nature of the lattice perturbation and on the new energy levels in the forbidden band that arise from this perturbation.

In the case of silicon and germanium the situation appears to be entirely different. The different widths of the bands may indeed account for the differences in the behavior of the material. If in the case of germanium one could assume that both acceptor and donor states are located below the middle of the forbidden band, then a \(p\)-type semiconductor should be obtained. In the case of silicon, where the band is considerably wider, when both acceptors and donors enter the middle of the forbidden band, regardless of whether \(p\)-type or \(n\)-type silicon is taken as the initial material, in the end silicon with high resistance should be obtained. The positions of the donor and acceptor levels determine the final position of the Fermi level*).

ANALYSIS OF THE RESULTS OF IRRADIATION WITH FAST NEUTRONS

During irradiation with fast neutrons it was observed that the conductivity of a \(p\)-type material increases smoothly, whereas the conductivity of an \(n\)-type material first decreases, reaches a minimum, and then again increases \(^{10-12}\). It was shown that on the initial section of the curve characterizing the course of the conductivity we are dealing with \(n\)-germanium; however, after the conductivity has reached its minimum, on the second branch of the curve we have \(p\)-type germanium \(^{12,27}\). To understand this behavior of the material, we suppose that vacant lattice sites act as acceptors, but that the activation energy of the latter is higher than the activation energy of acceptors introduced chemically. On the other hand, atoms in interstitial positions may be regarded as donors; however, near the filled band there are deeply lying donor levels. One might expect that acceptor states lying somewhat above the filled band and donor states lying just below the conduction band would neutralize one another. If such states are due to the displacement of atoms, they cannot be detected by electrical measurements.

*) See Addendum II.

The existence of such deeply lying states becomes understandable if one takes into account that, when vacancies arise and interstitial sites are filled, not only electrical but also elastic forces play a role. It should also be borne in mind that, in the case of germanium and silicon, the removal of an atom from a lattice site and its transition to an interstitial site is equivalent to the removal of more than one electron from one place (a vacancy) and the addition of more than one electron to another place (an interstitial) of the lattice. As a result, here we are dealing not with a hydrogen-like problem, but rather with a many-electron problem. The removal of the second and subsequent electrons requires an activation energy considerably exceeding the energy of hydrogen-like impurity terms, where only one electron is absent from, or is added to, the outer shell.

It is clear that, in the case of an \(n\)-type substance, both low- and high-lying acceptor levels will be filled by electrons from the conduction band. However, it is also clear that in a \(p\)-type substance the acceptor levels can become active only if they are filled by electrons excited from the filled band. If the acceptor states are situated too far from the filled band, they become active only at comparatively high temperatures. Similarly, if donor states are situated considerably below the conduction band, they can have an effect only at elevated temperatures. From this point of view it is clear that the resistance of \(n\)-germanium will increase more rapidly at the beginning of the irradiation process. It is also understandable that after the conductivity of the \(p\)-type semiconductor thus created again reaches a minimum, it will slowly increase.

This was shown quantitatively as a result of an analysis of the irradiation curves of \(p\)- and \(n\)-type semiconductors, carried out recently by Crawford\(^{28}\). We assume that in the initial stages of irradiation one may neglect the photoeffect due to the radiation, as well as the effect of “annealing” of the lattice. Likewise, the change in mobility may be neglected until a large number of lattice defects has been created. In this case the following relation holds:

\[ \frac{d\sigma}{d(n_n vt)_{\text{fast}}} = e b_e \frac{dn_e}{d(n_n vt)_{\text{fast}}}, \tag{9} \]

where \(n_e\) is the number of conduction electrons, \(b_e\) is the electron mobility, and \(\sigma\) is the conductivity. If it is assumed that all acceptor states lie much lower than the donor levels originally present, then each newly created state causes the removal of one electron from the conduction band. Then, for the initial portion of the curve, we obtain:

\[ \frac{d\sigma}{d(n_n vt)_{\text{fast}}} = e b_e \overline{z}_a . \tag{10} \]

In this equation \(\bar z_a\) is the average number of effective acceptors created by irradiation, depending to a considerable extent on the distribution of neutron energies. The initial value \(\bar z_a\) can be determined from the initial slope of the curve, if the initial values of the mobilities are known. Table IV gives the results of such an investigation for measurements carried out at various temperatures.

Table IV

Experimental determination of the number of acceptors per 1 neutron

Sample	Exposure temperature, °C	Initial number of carriers in \(\mathrm{cm}^{3}\)	Change in the number of effective carriers per 1 incident neutron in \(\mathrm{cm}^{3}\)
1	\(\sim 32\)	\(2.0\cdot 10^{15}\)	\(-1.7\)
2	30	\(4.3\cdot 10^{15}\)	\(-1.5\)
3	20	\(8.9\cdot 10^{15}\)	\(-3.5\)
4	\(\sim 32\)	\(2.8\cdot 10^{16}\)	\(-2.6\)
5	\(\sim 32\)	\(5.5\cdot 10^{16}\)	\(-5.2\)
6	\(-79\)	—	\(-4.3\)
7	\(-79\)	—	\(-3.1\)
8	\(-79\)	—	\(-3.2\)
9 *)	30	\(1.02\cdot 10^{16}\)	\(-3.9\)
10	0	\(2.5\cdot 10^{15}\)	\(+0.60\)
11	20	\(4.2\cdot 10^{15}\)	\(+0.70\)
12 **)	30	\(1.6\cdot 10^{15}\)	\(+0.77\)

) Samples 1 to 9 are of \(n\)-type.
*) Samples 10 to 12 are of \(p\)-type.

The number of carriers per incident neutron for an \(n\)-type substance is approximately 3; for a \(p\)-type substance it is from 0.6 to 0.8, depending on temperature. This is in agreement with the accepted scheme of energy levels. The number of carriers obtained for an \(n\)-type substance per incident neutron at the temperature of solid carbon dioxide is approximately the same. This indicates that, at first, our assumptions are confirmed.

This number is also in agreement with the theoretical value obtained by Ivens in analyzing the results of irradiating germanium with neutrons of energy \(2\) MeV. 135 displacements per scattered neutron correspond approximately to six displacements per incident neutron. Since Ivens took into account neither the effect of “healing” of the lattice nor recombination, this is in agreement with the experimental data.

Figure 4 shows the conductivity–irradiation curve for n-type germanium at 32°C. The curves obtained at the temperature of dry ice have the same form. The initial part of the curves is so rectilinear that it was possible to determine its slope accurately[^28]. Figure 5 illustrates the irradiation curve for p-type germanium with high resistivity. It is interesting to note that this

Fig. 4. Conductivity of n-type germanium at 32°C as a function of neutron flux. Initial characteristics: \(\sigma_0 = 0.77\ \Omega^{-1}\,\mathrm{cm}^{-1}\); \(b = 2920\ \mathrm{cm}^2/\mathrm{V\,sec}\), \(R = -3630\ \mathrm{cm}^3/\mathrm{coulomb}\); \(N_0 = 2.03 \cdot 10^{-15}\ \mathrm{cm}^{-3}\). Change in carrier concentration, if \(b\) is taken as constant,

\[ \frac{d\sigma}{d(n_n v t)} \approx -7.8 \cdot 10^{-16}\ \Omega^{-1}\,\mathrm{cm}^{-1}/\text{neutron}, \]

\[ -\frac{dn}{d(n_n v t)} \approx 1.7\ \text{carriers per incident neutron}. \]

curve is strictly rectilinear and that the changes in conductivity can be measured in a few minutes. This makes such an experiment very valuable for practical purposes. Additional information on the behavior of irradiated semiconductors can be obtained by considering the conditions corresponding to the minimum conductivity[^29]. In this case we see to what extent the high-intensity \(\beta\)- or \(\gamma\)-radiation given off by the reactor affects the behavior of the semiconductor near the minimum conductivity. If such transient effects occur, then we are certainly not dealing with a semiconductor in a state of thermal equilibrium.

The condition of thermal equilibrium between holes and electrons in the concentration range for which classical stati-

IRRADIATION OF SEMICONDUCTORS BY NUCLEONS

statistics, can be expressed as follows:

\[ n_e^0 n_h^0 = K(T)=\frac{4(2\pi \overline{m}^{*} kT)^3}{h^6}\exp\left(-\frac{\varepsilon^{*}}{KT}\right), \tag{11} \]

where \(\overline{m}^{*}\) is the mean geometric value of the effective masses of the carriers, \(\varepsilon^{*}\) is the width of the forbidden band, and \(K(T)\) is the equilibrium constant, dependent only on temperature. In our preceding considerations we used for \(K\) the value \(3.7\cdot 10^{27}\ \text{cm}^{-6}\) \((300^\circ\text{K})\).

The presence of the factor \(3\pi/8\) in the formula determining the Hall constant can be justified only in the case when we neglect scattering on impurities. This factor may change, and with it also changes the calculated value of the mobility. However, the electron mobility in the lattice, determined on the basis of drift measurements\({}^{30}\), is equal to \(b_e=3\cdot 300\ \text{cm}^2/\text{volt sec}\). If we introduce this mobility into the equation for the intrinsic conductivity, we obtain:

\[ \sigma_i = ne b_e\left(1+\frac{1}{\Lambda}\right), \tag{12} \]

where \(n\) is the number of intrinsic electrons and holes, and \(\Lambda\) is the ratio of the mobilities of electrons and holes, equal for germanium\({}^{31}\) to 1.5. Since \(K^{1/2}=n\), we find:

\[ K_{300^\circ\text{K}}=3.6\cdot 10^{36}\ \text{cm}^{-6}. \tag{13} \]

Fig. 5. Conductivity of \(p\)-type germanium at \(0^\circ\text{C}\) as a function of irradiation time (fast neutrons). Initial characteristics: \(\sigma_0=9.8\cdot 10^{-2}\ \Omega^{-1}\ \text{cm}^{-1}\); \(R=2.98\cdot 10^4\ \text{cm}^3/\text{coulomb}\), \(N_0=2.5\cdot 10^{14}\ \text{cm}^{-3}\); \(b_h=2450\ \text{cm}^2/\text{v sec}\);

\[ \frac{d\sigma}{d(n_nvt)_{\text{fast}}} = 2.36\cdot 10^{-16}\ \Omega^{-1}\ \text{cm}^{-1}/\text{neutron}; \]

\[ \frac{dn}{d(n_nvt)_{\text{fast}}} = 0.61 \]

acceptor per incident neutron.

Since the ratio of the mobility of electrons to the mobility of holes does not change with the addition of scattering centers of a new type, the equation for the conductivity can be written in the following form:

\[ \sigma=en_e^0 b_e+en_h^0 b_h =eb_h\left[n_e\Lambda+\frac{K(T)}{n_e}\right]. \tag{14} \]

Then the concentration of electrons at the minimum value of the conductivity is equal to

\[ n_e=\left(\frac{K}{A}\right)^{\frac12}, \tag{15} \]

and the minimum conductivity:

\[ \sigma_{\min}=2 e b_n (K A)^{\frac12}. \tag{16} \]

These relations and the experimental value of the minimum conductivity make it possible to determine the mobilities of the carriers in the semiconductor; the latter, in turn, can be compared with the initial mobility obtained as a result of measuring the primary Hall effect and the conductivity. Temporary ionization will increase the conductivity, as a result of which the calculated apparent value of the electron mobility will be significantly greater than the initial one. In reality, in the presence of additional scattering centers created by irradiation, the mobility at the minimum value of the conductivity must be less than at the beginning of the experiment, if the semiconductor is in a state of thermal equilibrium.

The calculated values of the mobility always indicate that the assumption of thermal equilibrium between the carriers for an irradiated semiconductor is not fulfilled[^29]. The circumstance that the observed minimum value of the conductivity is always much higher than the value of the intrinsic conductivity at the same temperature \((\sigma_i=0.0164)\) may be due either to the influence of the ionizing radiation and the transient effects arising in this case, or to inhomogeneity of the substance.

If the gradients of impurity concentration were normal to the direction of the current, then the exact value of the minimum conductivity could never be established. The system would then be equivalent to a series of parallel resistances, some of which would always possess a conductivity exceeding the minimum. On the other hand, if the concentration gradient is parallel to the direction of the current, then bombardment creates \((p-n)\)-boundaries between well-conducting \(p\)- and \(n\)-semiconductors; since, as has been shown by recent experiments, the \((p-n)\)-boundary acts as a counter (see below) and is not affected by \(\gamma\)- and \(\beta\)-radiation, it is clear that the expected value of the intrinsic minimum conductivity is not reached in this case, as a result of which the apparent mobility values become too high if they are calculated from the equation for \(\sigma_{\min}\).

The fact that the tangent of the angle of inclination of the conductivity–time curve after the formation of a \(p\)-type substance is significantly smaller than the initial tangent of the angle of inclination of the curve for the original \(n\)-type substance (less than one tenth of the initial value) is explained not

only by the presence of the distribution of acceptor states, which was mentioned above, but also by the recombination of lattice defects, which becomes appreciable only after a certain concentration has been reached. The annealing of lattice defects is considerably easier to study from the decay curve after irradiation than from the irradiation curve.

Fig. 6. Hall effect and resistance of \(n\)-type germanium bombarded by fast neutrons up to the attainment of the minimum value of the conductivity.

Dependence of \(R\) on \(10^3/T\):
\(I\)—initial,
\(II\)—after irradiation to minimum conductivity.
Dependence of \(\rho\) on \(10^3/T\):
\(III\)—initial,
\(IV\)—after irradiation to minimum conductivity.

Such experiments with neutrons are now being carried out*). It is possible, however, to predict to what extent the conductivity will change during irradiation as a function of recombination. The equation characterizing the formation of vacant sites should expediently be written in the following form:

\[ \frac{dN_v}{dt}=z_d-C_r\,(N_v-\Delta)\,N_i . \tag{17} \]

Here \(z_d\) is the rate of formation of defects, and the second term is the rate of recombination of atoms in interstices \(N_i\) and vacant sites \(N_v\).

*) The decay effect was observed earlier in experiments with \(\alpha\)-particles\({}^{32}\) of 20 MeV, and recently after irradiation of polonium by \(\alpha\)-particles.

The effective cross section of vacant sites here is equal to \((N_v-\Delta)\), taking into account the possibility of their coagulation. A preliminary investigation of the change in the conductivity of \(p\)-type germanium under irradiation by fast neutrons, carried out by Crawford\(^{34}\), indicates that annealing of the lattice is more likely a first-order process rather than a second-order one. Analysis of the \(\alpha\)-irradiation curve leads to analogous conclusions\(^{33}\).

Fig. 7. Conductivity of \(p\)-type silicon as a function of the flux of fast neutrons. (Note the effect of lattice annealing.) The rate of change of carrier concentration (the mobility is assumed constant)

\[ \frac{dn}{d(n_nvt)_{\text{fast}}} = \frac{1}{eb}\, \frac{d\sigma}{d(n_nvt)_{\text{fast}}}, \]

\[ \frac{d\sigma}{d(n_nvt)_{\text{fast}}} = -1.6\cdot 10^{-16}\ \Omega^{-1}\,\mathrm{cm}^{-1}/\text{neutron}, \]

\[ \frac{dn}{d(n_nvt)_{\text{fast}}} = -5\ \text{carriers per incident neutron}. \]

If the germanium specimen remains in the reactor until the minimum value of the conductivity is reached, one may expect that the tangent of the angle of inclination of the curve of the dependence of \(\ln \rho\) on \(\frac{1}{T}\) will approach its value for the case of intrinsic conductivity. Fig. 6 shows that this is indeed so, but that at low temperatures the magnitude of the activation energy, although considerably higher,

than is usually observed for germanium, but still below its intrinsic value (0.31 eV instead of 0.75 eV). This result should be compared with the results of transmutation experiments, when acceptors and donors are equally effective3.

In the case of other substances, the observed phenomena differ sharply from those described above for germanium. The case of silicon is of greatest interest. Since the latter has the same structure as germanium (a diamond lattice), and chemical impurities affect it in exactly the same way, one might have expected such effects to arise when it is bombarded. However, this is not so. In silicon, both of the \(n\)- and \(p\)-type, as a result of irradiation the resistance increases (see Fig. 7). In the case of a \(p\)-type substance this increase is considerably greater than in the case of an \(n\)-type substance3 (see Tables V and VI). This fact will become clear if we consider the results of experiments on heat treatment of silicon and optical measurements4,

Table V

Change in the resistance of \(p\)-type silicon under the action of fast neutrons

Time (sec.) . . . .	0	10,000	20,000	30,000	40,000
Resistance (ohms)	29.8	49.9	59.2	78.6	120
Time (sec.) . . . .	52,000	62,000	82,000	100,000	158,000
Resistance (ohms)	190	463	3,518	16,090	68,000

Table VI

Change in the resistance of \(n\)-type silicon under the action of fast neutrons

Time (sec.) . . . .	0	5,000	10,000	22,000	50,000
Resistance (ohms)	23.4	24.3	25.0	27.5	30.7
Time (sec.) . . . .	103,000	150,000	200,000	300,000	400,000
Resistance (ohms)	32.4	36.6	42.9	63.0	114.5

which indicate that the irradiated silicon has a certain number of deep-lying levels located between 0.7 and 0.9 ev in the middle of the forbidden band. If, in the middle of the band, where thermal ionization is very ineffective, donors and acceptors are formed in equal numbers, it is easy to see that an \(n\)-type conductor must give up all its conduction electrons to the acceptors. On the other hand, in the case of a \(p\)-type conductor, the electrons must fill the lower, not completely filled, band. In both cases the resistance increases.

Experiments show that, in \(p\)-type silicon, the resistance increases considerably faster than in \(n\)-type silicon. This means that the Fermi level is displaced in the direction of its limiting position in the upper half of the forbidden band, determined solely by defects.

The highest resistance so far observed in the bombardment of silicon is only \(10\,000\ \Omega\cdot\text{cm}\), which is almost 30 times smaller than the expected intrinsic value of the resistance.

X-ray diffraction studies\({}^{38}\) of irradiated silicon do not reveal new compounds, which are formed under irradiation only on the surface of the specimen. Powder diagrams of two irradiated specimens gave the following values of the lattice constant: \(5.4185 \pm 0.0005\ \text{Å}\) and \(5.4187 \pm 0.0005\ \text{Å}\), which is in agreement with the accepted values of this quantity. Irradiation of silicon by deuterons gives the same result as irradiation by neutrons: the resistance of \(p\)-type material increases considerably faster than the resistance of \(n\)-type material. Heat treatment in vacuum at \(450^\circ\text{C}\) restores the initial state of the specimen.

BOMBARDMENT OF GERMANIUM BY ELECTRONS

We have already mentioned that, when germanium is irradiated by electrons and \(\gamma\)-rays, one should expect the appearance of primary temporary effects. However, M. M. Mills pointed out that high-energy electrons may in this case also produce some permanent effects. In our experiments*) with electrons we observed only temporary effects, either because the electrons did not possess energy sufficient to cause displacements of atoms, or because the currents used were so weak that the number of displaced atoms was too small for the effect to become observable. If the order of magnitude of the recoil energy necessary for the complete displacement of an atom from a lattice site is estimated at 25 ev, then one may say that the 0.3-Mev electrons used in our earlier work possessed too little energy.

*) The experiments were carried out jointly with E. Klontz.

Using Mott’s theory \(^{39,40}\), we can estimate the cross section of this process, as well as the time required to produce the observed effect. Let an electron (rest mass \(m\)) with energy \(W_p\) and a given momentum collide with an atom of mass \(M_A\). Then the angle \(\theta\) between the direction of incidence and the recoil direction is related to the recoil energy \(W_r\) by the equation

\[ \cos \theta = \frac{ 1-\dfrac{W_r}{W_p} \left[ 1+\dfrac{(M_A-m)c^2}{W_p+2mc^2} \right] }{ \left[ 1-\dfrac{W_r}{W_p} \left( 1+\dfrac{W_p-W_r}{W_p+2mc^2} \right) \right]^{1/2} }. \tag{18} \]

For \(W_r \ll W_p\) and \(m \ll M_A\), this gives approximately

\[ \cos \theta = 1-\frac{M_Ac^2}{W_p+2mc^2} \left( \frac{W_r}{W_p} \right). \tag{19} \]

For a head-on collision \(\theta=\pi\), and approximately

\[ W_r= \frac{2W_pm}{M_A} \left( \frac{W_p}{mc^2}+2 \right). \tag{20} \]

If \(W_r\) is equal to \(25\ \text{eV}\), then the minimum value of the electron energy is \(0.54\ \text{MeV}\). Using the expressions obtained in Mott’s theory for the integral cross section

\[ \sigma_{\theta}^{180^\circ} = \pi Z^2 \left( \frac{e}{mc^2} \right) \left( \frac{1-\beta^2}{\beta^4} \right) \left\{ \operatorname{ctg}^2 \frac{\theta}{2} + 2\beta^2 \ln \sin \frac{\theta}{2} + \right. \]

\[ \left. + \frac{2\pi\beta Z}{137} \left( \sin \frac{\theta}{2} + \operatorname{cosec} \frac{\theta}{2} -2 \right) \right\} + \]

\[ + \text{terms of higher order relative to } \frac{Z}{137}, \tag{21} \]

one can estimate the currents and irradiation times required to produce the observed effects \(\left(\beta=\dfrac{v}{c}\right)\). For electrons with an energy of \(2\ \text{MeV}\) and a current density of \(1\ \mu\text{A}/\text{cm}^2\), the time required to obtain the observed effect is of the order of 10 minutes.

Idea of the experimental setup. In the case of high-resistance \(n\)-type germanium, the number of electrons subject to “neutralization” is small, and if \(n\)-type germanium is converted into \(p\)-type germanium, then this effect becomes observable over a measurable interval of time. Qualitatively, this can be detected by observing the change in the rectifier characteristic.

When bombarding a partially screened \(n\)-type germanium sample with electrons, one can obtain regions of \(p\)-type germanium, bounded—

protected regions of \(n\)-type germanium (the protected region). When measuring the rectifier characteristic it is easy to distinguish the two types of germanium, since the thickness of the layer that has undergone transformation is large in comparison with the barrier thickness. The latter condition is satisfied for energies of interest to us, since even \(0.3\)-MeV electrons, which do not cause permanent effects, penetrate into a germanium specimen to a depth of approximately \(0.02\ \mathrm{cm}\).

Experimental arrangement. A germanium specimen was irradiated at constant energies and current for various intervals of time. A lead shutter with a slit was placed above the specimen, so that only a small part of the specimen was irradiated at any one time. Irradiation of the specimen for different times and energies made it possible to determine approximately the position of the threshold of the effect. This also made it possible to ascertain whether some of the observed effects were due solely to heating. The specimen was then examined, and the shape of the rectification-characteristic curve for different parts of the specimen was observed on an oscilloscope. In order that each specimen could be irradiated several times, the specimens under investigation had to be as homogeneous as possible.

Fig. 8. Specimen holder and experimental arrangement for electron bombardment of germanium.

The experimental setup is shown in Fig. 8. In the course of the experiment, a large amount of energy had to be dissipated in a very small volume of the crystal. Therefore the specimen holder was made of a heavy copper strip, immersed in liquid air throughout the irradiation. The strip was drilled along its length and fitted with a spring-piston device, by means of which the specimen could remain in thermal contact with the block and be moved under the opening of the lead shutter. Thus the irradiated region could be accurately controlled, and after irradiation it was easy to investigate a definite portion of the specimen. The irradiation aperture was a square \(3\ \mathrm{mm}\) wide.

Each specimen was provided with a copper wire soldered across one of its ends to make contact during bombardment and measurements. In some experiments the temperature of the specimen was recorded. The rate of temperature rise ranged from \(0.07^\circ\mathrm{C}/\mathrm{sec}\) (for \(0.5\)-MeV electrons) to \(0.2^\circ\mathrm{C}/\mathrm{sec}\) (\(1\)-MeV), so that by the end of a 30-minute bombardment the temperature was approximately \(0^\circ\mathrm{C}\).

In some experiments a dense layer of frost covered the entire specimen holder. This made it necessary to raise the temperature of the copper

bands above zero in the intervals between irradiations, in order to make precise regulation possible.

Results of the experiments. In these experiments six germanium samples were subjected to irradiation. The results of the experiments are given in Table VII. We come to the conclusion that the threshold of the effect lies between 0.5

Table VII

Results of continuous bombardment of germanium of the p- and n-type
by high-energy electrons

Sample	Resistance in ohm cm	Bombardment No.	Time in sec.	Beam characteristic: energy in MeV	Beam characteristic: current in μa	Energy per sample in joules	Resistance of irradiated surface in ohm cm	Change in type of conductivity
n—1	11.6	1	200	2.0	10	400	∼ 26	Transition to p-type
n—1	11.6	2	2000	2.0	10	4000	32	Transition to p-type
n—2	4.75	1	300	1.5	10	450	44	From good to weak p-type conductivity
n—2	4.75	2	1200	1.0	10	1200	13	From good to weak p-type conductivity
n—2	4.75	3	2000	0.7	9	1260	5	From good to weak p-type conductivity
n—2	4.75	4	2400	0.5	6	720	Same	No change
n—3	16.3	1	42	2.0	10	84	40	Weak p-type conductivity
n—4	1.4	1	900	1.0	1.7 **)	1530	2.1	Weak p-type conductivity
n—5	5.0	1	900	0.7	1.0 **)	630	—	Weak p-type conductivity
n—5	5.0	2	1800	0.5	1.0 **)	900	—	No change
p—1	6.2	1	350	2.0	10	700	4.0	Resistance decreases

*) Current going to the sample, ∼0.1 of the total current.
**) Actual current going to the sample.

and 0.7 MeV. In order to establish its exact value, further experiments must be carried out. To determine whether the effect would be observed over the assumed interval of time, the specimen was irradiated for 42 sec at 2 MeV. The effect proved to be observable. It was to be expected that the resistance of the \(p\)-type specimen would decrease during bombardment, just as in the case of neutron irradiation, and this was confirmed experimentally.

These experiments show that, when germanium is irradiated with electrons possessing high energy, permanent effects can be observed.

These preliminary experiments also indicate that the minimum value of the energy necessary to remove a germanium atom from a lattice site is of the order of 30 eV, i.e., of the same order as that assumed in the existing literature. Immediately after irradiation of the specimen with 1-MeV and 0.7-MeV electrons for 15 min, a complete conversion of \(n\)-germanium into \(p\)-germanium takes place. During the time of measurement this effect partially disappears even at room temperature. Hence the note in Table VII: “weak \(p\)-type conductivity.” The fact that heating of the surrounding medium above room temperature is not the cause of the observed effect is quite obvious, since we subjected the specimens to electron bombardment from the electron gun of our electron-diffraction apparatus, the power of which was considerably higher than in the experiments with deuterons or electrons that did not produce any changes in conductivity.

The circumstance that bombardment of the substance for 30 minutes at 0.5 MeV produced no effect likewise indicates that the influence of the bombardment is not reducible to simple heat treatment. A considerably larger number of quantitative experiments will be required to decide the question of whether the observed effects are also caused by ionization processes or by “localized heating.” These processes may likewise lead to a lowering of the threshold for the permanent conversion of \(n\)-type germanium into \(p\)-type germanium. The fact that, in order to obtain a measurable electrical effect, it is sufficient to create a small number of impurity centers shows that, in the case of semiconductors, electrical measurements are a considerably more sensitive method for detecting structural changes than in the case of metals or other substances.

TIME PROCESSES

Many authors\(^{41—44}\) have investigated the question of the possibility of increasing the conductivity of a semiconductor by electron or ionizing irradiation. In connection with our experiments on bombardment by nucleons, we irradiated germanium specimens with electrons of energy 0.3 MeV from our Van de Graaff generator. The specimens

germanium of thickness \(0.02\ \text{cm}\) were “transparent” for such irradiation; thus we observed bulk effects.

An \(n\)-type germanium sample with a resistivity of \(18\ \Omega\cdot\text{cm}\) was placed in an evacuated chamber attached at one end to the discharge column of an electrical generator. The sample was irradiated with electrons of energy \(340\ \text{keV}\) in microsecond pulses produced by a thyratron circuit. The pulse frequency was \(200\) per second, and the current was \(300\ \mu\text{A}\) per pulse. A resistance of \(2200\ \Omega\) and a DC voltage were connected in series with the sample. Changes in time of the current passing through the sample and of the voltage across it were observed with an oscilloscope whose sweep was synchronized with the pulse generator. From these measurements the resistance was calculated as a function of time; the results indicate a drop in resistance from \(1460\) to \(400\ \Omega\) during each pulse. Subsequently, complete recovery of the resistance to its initial value was observed; however, this process did not occur instantaneously, but in approximately \(10\ \mu\text{s}\) (Fig. 9).

Fig. 9. Change in resistance in \(n\)-type germanium under pulsed electron irradiation. Pulse characteristics: duration \(=1\ \mu\text{s}\); frequency \(=200/\text{s}\); peak \(=300\ \mu\text{A}\); energy \(=230\ \text{keV}\).

\((p-n)\) barriers as counters. In some of our experiments on producing \((p-n)\) junctions in germanium by irradiation with deuterons and \(\alpha\)-particles, it was found that after bombardment the change in the junction characteristics ceases. It was suggested that this may be due to radioactivity induced in the substance, and that such a \((p-n)\) junction can act as a counter for ionizing radiation\(^{45}\). It turned out that our assumptions were correct.

The author investigated both natural junctions formed in a melt and artificial junctions produced by bombardment with deuterons. Natural \((p-n)\) junctions have a thickness of \(1\cdot 0.3\cdot 0.1\ \text{cm}\), and artificial junctions \(0.5\cdot 0.2\cdot 0.02\ \text{cm}\). The behavior of natural and artificial junctions is essentially identical, apart from differences caused by the difference in their thickness, namely, their unequal transparency with respect to \(\beta\)- and \(\gamma\)-radiation. The effective area of the counter is of the order of \(10^{-4}\ \text{cm}^2\). The pulse rise time is less than \(0.05\ \mu\text{s}\). The maximum pulse height is of the order of \(2\ \text{mV}\).

Since the \((p-n)\)-junction is very sensitive to light, it is quite essential that it be placed in an absolutely light-tight thermostat. Its properties were studied at various temperatures, from room temperature down to the temperature of liquid nitrogen. In the region of room temperature, the temperature measurements were carried out with an accuracy of \(\pm 0.05^\circ\)C. The load resistance was \(10^5\ \Omega\) (with a total voltage drop of \(100\ \mathrm{V}\)), and the bias voltage across the junction was \(10\ \mathrm{V}\), producing a bias current of the order of \(1\ \mathrm{mA}\). At the temperature of liquid air the resistance of the junction is of the order of \(100\ \mathrm{M}\Omega\). The \((p-n)\)-junction is very sensitive to radiation at room temperature; therefore it must be carefully shielded. It is sensitive to \(\alpha\)-, \(\beta\)-, and \(\gamma\)-radiation. After irradiation of the \((p-n)\)-junction with more than \(10^7\) \(\alpha\)-particles, the maximum of the pulse-height distribution slowly shifts toward lower values. The \((p-n)\)-junction is stable with respect to small changes in temperature, but the maximum of the pulse-height distribution is very sensitive to slight temperature fluctuations. The pulse-height distribution is typical of crystalline counters with a small cross section. It has a very sharp maximum for \(\alpha\)-particles; however, this maximum is broadened in the case of \(\beta\)- and \(\gamma\)-particles. The \((p-n)\)-junction is the most interesting object for studying the mechanism of operation of a crystalline counter. In this case one can vary a large number of parameters, such as resistance, dimensions, etc., which change comparatively little in ionic crystalline counters. The number of carriers can be varied at will either by regulating the temperature of the specimen or by illuminating it with light. In addition, we hope, by means of a more careful and quantitative study of the operation of the \((p-n)\)-junction as a counter, to approach more deeply the study of the properties both of the junction itself and of germanium subjected to ionizing radiation.

CONCLUSIONS

1. Nuclear transmutations caused in germanium by slow neutrons lead to the formation of an excess of \(p\)-type germanium; and the values of the cross sections determined for the separated isotopes of germanium indicate that the amount of the expected excess of \(p\)-type impurities should be equal to \(2\%\) of the total flux of slow neutrons. Under irradiation with slow neutrons, approximately three times more \(p\)-type impurities (gallium) than \(n\)-type impurities (arsenic) were obtained.

Comparing the results obtained for the integral flux with the results of measuring the Hall effect after irradiation and heat treatment, it can be shown that each impurity center liberates one carrier and that \(n\)- and \(p\)-type impurity centers are equally effective. By irradiation for definite intervals of time and by gradually increasing the intensity

Irradiation of Semiconductors by Nucleons

irradiation it is possible to create a substance possessing the properties of intrinsic semiconductors, owing to the existence of equilibrium between two types of impurities.

1. Germanium of the \(n\)-type is converted into germanium of the \(p\)-type, and the conductivity of the latter increases upon irradiation with \(\alpha\)-particles, deuterons, neutrons, and fast electrons. The use of well-focused beams of charged particles makes it possible to obtain \((p-n)\)-boundaries located at definite places in the specimen.

The resistance of an \(n\)-type substance reaches a maximum when the electron conductivities are “neutralized”; when conversion into \(p\)-germanium takes place, the resistance again decreases. The number of displaced germanium atoms can be calculated both for bombardment by charged particles and for irradiation by neutrons; however, the true number of carriers detected is considerably smaller than the number of such displacements. This circumstance is due to the annealing effect of the lattice and also to the fact that the acceptor and donor levels created by lattice defects partially compensate one another.

Analysis of the neutron-irradiation curves shows that in \(n\)-type germanium, upon irradiation with fast neutrons, each incident neutron removes three carriers, whereas in the case of a \(p\)-type substance, depending on the temperature, approximately \(0.7\) carrier is added per incident neutron. This suggests that the lattice defects produced by bombardment act as donors and acceptors, possessing higher activation energies and distributed over a broader range of energies than chemical impurities.

It can be shown that the number of nuclear transmutations caused by deuterons cannot be regarded as sufficient to explain the large changes in conductivity that are observed upon irradiation with deuterons of energy \(10\) MeV. Heat treatment at a temperature of \(450^\circ\)C restores the initial properties of the irradiated semiconductor, as has already been mentioned above.

1. Irradiation with electrons of energy from \(0.7\) MeV and higher produces in germanium the same permanent effects as irradiation with heavy particles: conversion of \(n\)-type germanium into \(p\)-type germanium. From the lower limit of the energy necessary to produce this effect, one can calculate the minimum energy required for the complete removal of a germanium atom from the lattice site it occupies. This energy proved to be of the order of \(30\) eV. Electrons with energy \(0.5\) MeV do not produce such effects; this indicates that heating of the surrounding medium cannot be responsible for the phenomena observed at \(0.7\) MeV. This is also indicated by experiments with prolonged irradiation by electrons possessing a considerably lower energy (\(50\) keV).

1. By using electrons with energy below \(0.7\ \text{MeV}\) and the appropriate oscillographic equipment, it can be shown that such irradiation of germanium creates conductivity pulses and decreases the resistance by approximately a factor of 10; these conductivity pulses, however, last no more than \(10\ \mu\text{sec}\).

Transient effects were also observed in the case when natural and artificial \((p-n)\)-junctions in germanium were subjected to ionizing irradiation. One can make a crystal counter which, at room temperature, responds to \(\alpha\)-, \(\beta\)-, and \(\gamma\)-radiation. In this case the rise time of the pulse is less than \(0.05\ \mu\text{sec}\), the maximum pulse height is of the order of \(2\ \text{mV}\), and its displacement does not exceed \(10\ \text{V}\).

ADDENDUM

The effect of irradiation on a classical semiconductor in thermal equilibrium*)

The behavior of germanium under neutron bombardment is very similar to the behavior of lead sulfide when acted upon by oxygen. Just as in the case of irradiated germanium, in lead sulfide we observe transitions from \(n\)-type to \(p\)-type, and an increase of resistance up to a certain maximum, followed by a decrease of resistance when we obtain a \(p\)-type substance as a result of introducing a sufficient amount of oxygen. Therefore the following general considerations concerning the conductivity, Hall effect, and thermoelectromotive force of a semiconductor in thermal equilibrium as functions of irradiation are of interest.

If it is assumed that classical statistics are applicable**) and that the semiconductor is in thermal equilibrium, then for the conductivity one may write the expression

\[ \sigma = n_e e b_e + n_h e b_h = e b_h (n_e \Lambda + n_h) = e b_h \left(\frac{\Lambda K}{n_h} + n_h\right), \tag{22} \]

where \(\Lambda\) is again the ratio of the electron mobility to the hole mobility, and \(n_e n_h = K\) is the equilibrium constant at the given temperature. In an analogous manner, for the Hall coefficient one may write

\[ R = \frac{3\pi}{8e}\,\frac{n_h b_h^2 - n_e b_e^2}{(n_h b_h + n_e b_e)^2} = \frac{3\pi}{8}\,\frac{(n_h - \Lambda^2 n_e)}{(n_h - \Lambda n_e)^2} = \frac{3\pi n_h (n_h^2 - \Lambda^2 K)}{8e (n_h^2 + \Lambda K)^2}, \tag{23} \]

*) V. A. Johnson and K. Lark-Horovitz.

**) At a temperature of \(300^\circ\text{K}\) the Fermi level lies below the conduction band (or above the filled band) by \(3kT\) (or farther, for all carrier concentrations that are less than \(10^{18}\) per \(\text{cm}^3\)). This condition ensures the applicability of classical statistics of the type considered here.

and for the thermoelectromotive force

\[ -\frac{dE}{dT} = -\frac{k}{e} \left[ 2(n_e\Lambda-n_h) - n_e\Lambda\ln \frac{n_e h^3}{2(2\pi m_e^* kT)^{3/2}} + n_h\ln \frac{n_e h^3}{2(2\pi m_h^* kT)^{3/2}} \right](n_e\Lambda+n_h)^{-1}. \tag{24} \]

The conductivity reaches its minimum value when

\[ n_h=\Lambda n_e=(\Lambda K)^{1/2}. \tag{25} \]

The minimum value of the conductivity is determined by the formula

\[ \sigma_{\min}=2eb_h(\Lambda K)^{1/2}. \tag{26} \]

For comparison, let us note that the intrinsic conductivity at the same temperature is equal to

\[ \sigma_i=(1+\Lambda)eb_h K^{1/2} \tag{27} \]

and, thus*)

\[ \frac{\sigma_{\min}}{\sigma_i}=\frac{2\Lambda^{1/2}}{1+\Lambda}. \tag{28} \]

Thus: \(\sigma_{\min}=\sigma_i\), if \(\Lambda=1\); \(\sigma_{\min}<\sigma_i\), if \(\Lambda>1\), and \(\sigma_{\min}>\sigma_i\), if \(\Lambda<1\).

Now let us consider the value of the Hall coefficient at the moment when \(\sigma=\sigma_{\min}\). If we put \(n_h=(\Lambda K)^{1/2}\) and \(n_e=\left(\dfrac{K}{\Lambda}\right)^{1/2}\), we obtain:

\[ R'=\frac{3\pi}{32e}\frac{1-\Lambda}{(\Lambda K)^{1/2}}. \tag{29} \]

Thus, \(R'\) is negative, equal to zero, or positive depending on whether \(\Lambda\) is greater than, equal to, or less than unity. In the case of germanium the mobility ratio \(\Lambda>1\), and therefore, when the conductivity reaches a minimum, the sign of the Hall effect

*) This relation is valid if the values of \(b_h\) are identical for \(\sigma_{\min}\) and \(\sigma_i\). Irradiation that creates additional scattering centers tends to lower \(b_h\) for \(\sigma_{\min}\); as a result, the ratio \(\dfrac{\sigma_{\min}}{\sigma_i}\) becomes smaller than indicated. This effect depends on the initial concentration of free electrons due to impurities.

Here no noticeable decrease of \(b_h\) is observed for \(n<10^{14}\ \text{cm}^{-3}\), about a 1% decrease for \(n=10^{15}\ \text{cm}^{-3}\), and 10% for \(n=10^{16}\ \text{cm}^{-3}\).

nevertheless turns out to be negative, even if the number of holes is greater than the number of electrons. The Hall coefficient passes through 0 in order then to become positive, when \(n_h=\Lambda K^{1/2}\) and \(n_e=K^{1/2}/\Lambda\).

In exactly the same way, investigating the thermoelectromotive force at \(\sigma=\sigma_{\min}\), we find:

\[ \left(\frac{dE}{dT}\right)'=\left(-\frac{k}{2e}\right)\ln\left(\frac{\Lambda m_e^{3/2}}{m_h^{3/2}}\right). \tag{30} \]

This leads to the formula

\[ \left(\frac{dE}{dT}\right)=\left(-\frac{k}{2e}\right)\ln\frac{m_h}{m_e}, \tag{31} \]

if the ratio\(^{46}\) of the mobilities \(\Lambda\) is taken equal to \(\left(\dfrac{m_h}{m_e}\right)^{5/2}\). Thus, we find that \(\dfrac{dE}{dT}\) is negative, becomes zero, or is positive depending on whether \(m_h\) is greater than, equal to, or less than \(m_e\); the latter conditions correspond respectively to \(\Lambda>1\), \(\Lambda=0\), \(\Lambda<1\). If it is assumed that the semiconductor remains in thermal equilibrium, one can study in detail the Hall effect and the change in conductivity as functions of irradiation. Calculations of this kind were carried out by Putley\(^{47}\) and Freim\(^{48}\) in simplified form in connection with the study of the properties of lead sulfide. Möser\(^{49}\) carried out detailed investigations of the position of the Fermi level also for the case of lead sulfide. It is assumed that in experiments with an initial \(n\)-type germanium specimen, bombardment gives rise to a density of defects proportional to the flux, and that electrons from the conduction band pass to acceptor levels associated with lattice defects; this leads to a decrease in the number of free electrons practically to zero and to the subsequent formation of holes in the filled band, i.e., to the conversion of \(n\)-germanium into \(p\)-germanium.

The condition of thermal equilibrium\(^{50}\)

\[ n_h n_e=K=3.61\cdot10^{26}\ \mathrm{cm}^{-6}\quad \text{at }300^\circ \mathrm{K}, \tag{32} \]

Before the beginning of irradiation \(n_e=N+n\) and \(n_h=n\), where \(N\) is the initial electron density due to impurities, and \(n\) is the corresponding density of intrinsic electrons or holes. Let \(N'\) be the density of defects due to irradiation and subsequent nuclear transmutation. Then, since \(N'\leq N\), one may write

\[ n_e=N-N'+n,\quad n_h=n. \tag{33} \]

Let us note that \(n\) changes with the change of \(N\) and \(N'\) in such a way that \(n_e n_h=K\) throughout the whole time. When \(N'\geq N\), we set

\[ n_h=n+N'-N;\quad n_e=n. \tag{34} \]

At the beginning of irradiation, since \(N' < N\), the Hall coefficient is determined by the formula

\[ R=\frac{r_c}{e}\, \frac{ \left[ (1-\Lambda^2)\left(K+\frac{N_1^2}{4}\right)^{1/2} -(1+\Lambda^2)\frac{N_1}{2} \right] }{ \left[ (1+\Lambda)\left(K+\frac{N_1^2}{4}\right)^{1/2} +(\Lambda-1)\frac{N_1}{2} \right]^2 }, \tag{35} \]

where \(N_1=N-N'\) and \(r_c\) is a numerical coefficient of the order of unity, whose exact value depends on the nature of the scattering of the carriers. When \(N'=N\), we find that

\[ R=\left(-\frac{r_c}{eK^{1/2}}\right)\frac{\Lambda-1}{\Lambda+1}. \tag{36} \]

Thus, the Hall coefficient is always negative if \(\Lambda>1\). When \(N'>N\), the Hall coefficient acquires the value

\[ R=\frac{r_c}{e}\, \frac{ \left[ (1-\Lambda^2)\left(K+\frac{N_2^2}{4}\right)^{1/2} +(1+\Lambda^2)\frac{N_2}{2} \right] }{ \left[ (1+\Lambda)\left(K+\frac{N_2^2}{4}\right)^{1/2} -(\Lambda-1)\frac{N_2}{2} \right]^2 }, \tag{37} \]

where \(N_2=N'-N\).

The corresponding expressions for the conductivity have the form

\[ \sigma=eb_e\left[ \left(1+\frac{1}{\Lambda}\right) \left(K+\frac{N_1^2}{4}\right)^{1/2} -\left(1-\frac{1}{\Lambda}\right)\frac{N_1}{2} \right] \tag{38} \]

and

\[ \sigma=eb_e\left[ \left(1+\frac{1}{\Lambda}\right) \left(K+\frac{N_2^2}{4}\right)^{1/2} -\left(1-\frac{1}{\Lambda}\right)\frac{N_2}{2} \right]. \tag{39} \]

Fig. 10 shows the dependence of \(R\) and \(\sigma\) on \(N'\), calculated for a germanium sample at \(N=10^{16}\ \mathrm{cm}^{-3}\). In this calculation \(r_c\) was taken equal to 1, \(\Lambda=1.50\), and \(b_e\) equal to \(2800\ \mathrm{cm}^2/\mathrm{V\,sec}\).

The expressions for \(R\) were differentiated in order to find the maximum and minimum. The positions of the latter depend on \(\Lambda\) and \(K\). The results of the calculations are as follows:

\[ R=R^-_{\min}\quad \text{for}\quad N_1=1.245\cdot K^{1/2}=2.36\cdot 10^{13}/\mathrm{cm}^3. \]

and

\[ R^-_{\min}=-\frac{0.330\,r}{eK^{1/2}}\simeq -108\,000\ \mathrm{cm}^3/\mathrm{coulomb}. \]

In exactly the same way,

\[ R = R_{\max}^{+} \quad \text{for} \quad N_2 = 3.005 \cdot K^{1/2} = 5.71 \cdot 10^{13}/\text{cm}^3 \]

and

\[ R_{\max}^{+} = \frac{0.1858\,r}{eK_i^{1/2}} \approx 61\,000\ \text{cm}^3/\text{coulomb}. \]

Thus, as irradiation proceeds, we observe that the negative Hall coefficient increases in magnitude up to a certain maximum, while the conductivity decreases; thereafter the negative Hall constant

Fig. 10. Hall coefficient and conductivity of germanium in a state of thermal equilibrium as a function of the concentration of impurity centers created by neutron irradiation.

(Labels in the figure: Hall coefficient; conductivity; concentration of impurity centers \((10^{15}/\text{cm}^3)\); units \(\text{cm}^3/\text{coulomb}\) and \(\text{ohm}^{-1}\text{cm}^{-1}\).)

of Hall begins to decrease in magnitude, while the conductivity continues to decrease. Then the conductivity reaches its minimum and begins to increase before the Hall coefficient reaches zero; thereafter the conductivity continues to increase, while the Hall coefficient passes through a positive maximum and then falls.

ADDENDUM II

Fermi levels under irradiation of semiconductors*)

In studying the question of the position of the Fermi level in irradiated semiconductors, we shall assume that all donor states created by irradiation have the same activation energy and that all acceptor states likewise have the same energy.

*) H. M. James and J. W. Leman.

The adopted notation is illustrated by Fig. 11. The zero of energy is taken to be the level located in the middle of the forbidden band, whose width is \(\varepsilon^*\); thus, the edge of the conduction band corresponds to the energy \(+\dfrac{\varepsilon^*}{2}\), and the edge of the filled band to the energy \(-\dfrac{\varepsilon^*}{2}\). It is further assumed that the density \(N_+\) of donor levels with energy \(\varepsilon_D\) and the density \(N_-\) of acceptor levels with energy \(\varepsilon_A\), introduced chemically, remains constant. Irradiation creates new donor levels with energy \(\varepsilon_\alpha\) and acceptor levels with energy \(\varepsilon_\beta\).

Fig. 11. Band diagram for a semiconductor subjected to bombardment. The zero of energy is taken to be the level located in the middle of the forbidden band.

The total concentration of donors and acceptors created by bombardment is equal to \(N'\); the fraction \(f_\alpha\) of this concentration consists of donors, and the fraction \(f_\beta = 1 - f_\alpha\) consists of acceptors.

If \(\mu\) is the Fermi level, then the probability of ionization of a donor of \(n\)-type is

\[ P^+(w;\mu)= \frac{1}{1+\exp\left[-\frac{w-\mu}{kT}\right]}, \tag{40} \]

and the probability of ionization of an acceptor of \(p\)-type is

\[ P^-(w;\mu)= \frac{1}{1+\exp\left[\frac{w-\mu}{kT}\right]}, \tag{41} \]

where in both cases \(w\) is the energy of the corresponding level.

The concentration of electrons in the conduction band is

\[ n_e(\mu)=\frac{4\pi(2m_e^*)^{3/2}}{h^3} \int_0^\infty \frac{w^{1/2}\,dw}{1+\exp\left[\dfrac{w+\frac{\varepsilon^*}{2}-\mu}{kT}\right]}, \tag{42} \]

and the concentration of holes in the filled band is

\[ n_h(\mu)=\frac{4\pi(2m_h^*)^{3/2}}{h^3} \int_0^\infty \frac{w^{1/2}\,dw}{1+\exp\left[\dfrac{w+\frac{\varepsilon^*}{2}+\mu}{kT}\right]}, \tag{43} \]

where \(m_e^*\) and \(m_h^*\) are the effective masses, respectively, of electrons and holes.

The position of the \(\mu\)-level is determined by the condition of electrical neutrality of the crystal

\[ n_h(\mu)-n_e(\mu)+N_+P^+(\varepsilon_D;\mu)-N_-P^-(\varepsilon_A;\mu)+ N'\left[f_\alpha P^+(\varepsilon_\alpha;\mu)-f_\beta P^-(\varepsilon_\beta;\mu)\right]=0. \tag{44} \]

Here three special models will be investigated; in all cases we put \(\varepsilon^*=0.75\ \text{eV}\), which approximately corresponds to the width of the forbidden band of germanium, and \(m_e^*=m_h^*=m\). The temperature is equal to \(20^\circ\text{C}\).

Model I. It is assumed that all the chemical impurities present are completely ionized, so that the condition determining the position of \(\mu\) takes the form

\[ n_e(\mu)-n_h(\mu)+N_- - N_+ = N'\left[f_\alpha P^+(\varepsilon_\alpha;\mu)-f_\beta P^-(\varepsilon_\beta;\mu)\right]. \tag{45} \]

It is further assumed that in each case there is a chemical impurity of only one type. Irradiation creates levels \(\varepsilon_\alpha=0.175\ \text{eV}\) and \(\varepsilon_\beta=-0.175\ \text{eV}\). The values of \(f_\alpha\) considered are \(0.4\), \(0.5\), and \(0.6\), which corresponds to the presence of an excess of acceptors, equality of the numbers of acceptors and donors, and the presence of an excess of donors.

Fig. 12 gives the dependence of \(\mu\) on \(N'\) for an initial specimen of \(p\)-type. The corresponding curves for an \(n\)-type substance are obtained as a result of mirror reflection of these curves with respect to the zero level and replacement of \(N_-\) by \(N_+\) and \(f_\alpha\) by \(1-f_\alpha\).

If \(f_\alpha=0.6\), then to each pair of acceptors there correspond three donor levels, and the initial \(p\)-type substance, upon bombardment, is transformed into an \(n\)-type substance. The excess of the number of donors over the number of acceptors is equal to \(N'/5\). We note that in the case when \(N'/5\) is equal to the initial excess of acceptors, the \(\mu\)-level each time passes thro—

through \(w=0\). The limiting position of the \(\mu\)-level at large values of \(N'\), \(0.1575\) eV, is completely determined by the character of the additional levels that have appeared as a result of irradiation, and corresponds to an ionization probability of donors of the order of \(2/3\).

If \(f_{\alpha}=f_{\beta}=0.5\), then, owing to the symmetry of this model, the \(\mu\)-level approaches the middle of the band. The resistance of the specimen will then

Fig. 12. Position of the Fermi level as a function of \(N'\) for a substance which, before bombardment, was of \(p\)-type.

increase mainly during irradiation, independently of whether we initially had material of \(n\)- or \(p\)-type. If \(f_{\alpha}=0.4\), then irradiation increases the excess of acceptors, and the substance will still be of \(p\)-type. It is interesting to note that nevertheless the \(\mu\)-level will rise and the resistance increase if the initial concentration of completely ionized acceptor atoms is high. This occurs because donors fill a large part of the sites in the “filled” band; the greater part of the holes in the irradiated substance is captured by acceptor levels that are difficult to ionize.

Model II. Here it is assumed that the activation energy of the acceptors introduced chemically is equal to zero; \(\varepsilon_D=-\varepsilon_A=0.375\) eV. (Let us note that this does not imply the presence of complete ionization.) It is again assumed that in each case there is a chemical impurity of only one type. It is further assumed that irradiation creates equal concentrations of donor and acceptor levels (\(f_{\alpha}=f_{\beta}=0.5\)) with energies \(\varepsilon_{\alpha}=-0.225\) eV and \(\varepsilon_{\beta}=-0.325\) eV, respectively.

Fig. 13 depicts the dependence of \(\mu\) on \(N'/2\) (the number of additional donor or acceptor levels) for various initial concentrations of chemical impurities. In all cases, as \(N'\) increases, the \(\mu\)-level tends to the limit \(w=-0.275\ \text{eV}\).

The behavior of this model is analogous to the behavior of a germanium specimen: initially \(n\)-germanium turns into \(p\)-germanium, and the resistance of \(p\)-germanium at first decreases if the concentration of chemical acceptors is small. In the case of very contaminated \(p\)-type germanium, on the other hand, the resistance increases during irradiation. It should be expected that in all cases where the mobilities are the same, the value of the resistance tends to one and the same limit.

Fig. 13. Position of the Fermi level as a function of the concentration of additional donor and acceptor levels for various initial concentrations.

Model III. This model is intended for investigating the behavior of germanium containing impurities created by nuclear transmutations, under the condition that atoms located in interstitial sites and vacant sites have been removed by heat treatment.

It is assumed that the activation energy of the initial impurities of both \(n\)- and \(p\)-type is equal to zero; therefore the \(n\)- and \(p\)-type impurities created by irradiation are present in the ratio \(1:3\). The position of the \(\mu\)-level is determined by the equation

\[ n_e(\mu)-n_h(\mu)+\left[N_-+\frac{3}{4}N'\right]p^+(-0.375;\mu)- \]

\[ -\left[N_++\frac{1}{4}N'\right]p^-(+0.375;\mu)=0. \tag{46} \]

Fig. 14 gives the dependence of \(\mu\) on \(N'/2\), i.e., on the excess of acceptors over donors. It should again be noted that an \(n\)-type substance is converted into a \(p\)-type substance, while the resistance of the \(p\)-type substance decreases.

Fig. 14. Relation between the position of the Fermi level and the excess of acceptors over donors caused by nuclear transmutation.

In all cases the limiting position of the \(\mu\)-level is \( -0.3925\) ev below the edge of the filled band.

CITED LITERATURE

1. K. Lark-Horovitz, Nat. Def. Res. Comm. Rep. No. 14—585, pp. 7—57, May 1942—November 1945.
2. J. Bardeen and G. L. Pearson, Phys. Rev. 75, 865 (1949).
3. J. H. Scaff, H. C. Theuerer and E. E. Schumacher, Trans. Amer. Inst. Min. (metal) Eng. 185, 383 (1949).
4. J. Bardeen and W. Brattain, Phys. Rev. 75, 1216 (1949).
5. R. Bray, Purdue, Progr. Rep. Feb. 1946—Feb. 1949, p. 70.
6. Ibid., Sept. 1949—November 1949, p. 62.
7. E. Klontz and K. Lark-Horovitz, Progr. Rep., Contract No. W 36—039—32020 (Signal Corps), August—October 1948, p. 41.
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8. E. Bleuler and D. J. Tendam, Progr. Rep. Nov. 1947—Jan. 1948.
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1. The experiments were carried out jointly with I. Cleland, I. K. Crawford, and I. K. Pitt. ↩

2. Visible reference marker in the source text. ↩