For the first time, a quantitative model of the Curie–Weiss behavior of a low-temperature paramagnetic susceptibility of electrically neutral donors in *n*-type diamagnetic covalent semiconductors is proposed. The exchange interaction between nearest two neutral donors was calculated with the use of the Heitler–London model. In this model, we take into account the change in the thermal ionization energy of donors due to the shift of the bottom of the conduction band to the bandgap with doping and compensation. The energy of the exchange spin–spin interaction between electrons localized on donors is calculated as a function of the donor concentration and the degree of their compensation by acceptors. The broadening of the donor band due to the Coulomb interaction of the nearest impurity ions was taken into account. We considered crystals of *n*-type germanium doped with arsenic up to the concentration close to the insulator–metal phase transition (Mott transition) and compensated with gallium. The compensation ratio *K* is the ratio of the concentration of compensating acceptors *KN* to the concentration of doping donors *N*. The model predicts a change in the sign of the Curie–Weiss temperature from minus to plus (a transition from the antiferromagnetic to ferromagnetic local ordering of electron spins on donors) for *K* ≈ 0.15–0.3, reaching its maximum positive values of ≈1.3 K for *K* ≈ 0.5 with the following decrease (a transition to paramagnetism) for *K* > 0.85. The calculated behavior of the paramagnetic susceptibility of donors is consistent with the experimental data for compensated *n*-Ge:As,Ga samples close to the Mott transition.

## I. INTRODUCTION

Despite the more than half a century history of study of the electron spin resonance (ESR) of hydrogen-like (shallow) impurities in covalent semiconductor crystals, beginning with the work of Wilson,^{1} this subject still attracts much attention. In the last decade, a number of studies^{2–5} were devoted to the low-temperature^{6} (down to units of kelvin) magnetic susceptibility of crystalline semiconductors doped with hydrogen-like donors under the conditions of interaction of electron spins localized on donors, in particular on the insulator side of the concentration insulator–metal phase transition (Mott transition). The impurity diamagnetic susceptibility of a doped semiconductor close to this transition was experimentally determined for the first time^{4} using a comparative study by the methods of superconducting quantum interference device (SQUID) magnetometry and ESR. It has been shown by ESR spectroscopy (see, e.g., Ref. 2 and an earlier work, Ref. 7) that in semiconductor crystals close to the Mott transition electrons with oppositely directed spins, which are weakly localized on the nearest impurities, may form the so-called antiferromagnetic spin glass. (On the other hand, the existence of the magnetic ordering in semiconductor materials and structures doped with shallow impurities is also supported experimentally with the use of methods different from ESR.^{8,9})

As was found using the ESR technique,^{3,4} in the compensated *n*-Ge:As,Ga, which is close to the insulator–metal phase transition, the temperature decrease (below a few kelvin) leads to a transition from the Pauli paramagnetism^{10} to the Curie paramagnetism, accompanied by an anomalous increase in the paramagnetic susceptibility and *g*-factor, which is characteristic of a magnetic transition. A qualitative model of this phenomenon was proposed^{5} on the basis of the effect of localization of electron spins in the Coulomb gap, which arises due to the Coulomb blockade of electrons in a “metalized” impurity band under the field of compensating impurities at low temperatures corresponding to the thermal energy smaller than the gap width. Nevertheless, the question of the theoretical substantiation of possible types of magnetic ordering of shallow impurities in covalent diamagnetic semiconductors is still lacking.

In this paper, we propose a model for the transition from the antiferromagnetic ordering of electron spins of shallow paramagnetic impurities (donors) to the ferromagnetic ordering. This transition is affected by internal (doping level and degree of compensation) and external (temperature and magnetic field) factors. The model is based on the phenomenological theory of the molecular field (Weiss field)^{11} for low temperatures, when the concentration of electrons in the states of the donor band is much higher than the concentration of electrons in the conduction band (*c*-band). Such a consideration is applicable for not too high doping levels (when the band of donor states is still separated by the gap from the bottom of the *c*-band) and the smallness of the average kinetic energy of a *c*-band electron compared to the average Coulomb energy of its attraction to the ionized donor.^{12}

Let us consider a crystalline *n*-type diamagnetic covalent semiconductor with hydrogen-like impurities. The concentration of donors in the charge states (0) and (+1) is *N* = *N*_{0} + *N*_{+1}, and the concentration of acceptors, which are all in the charge state (−1), is *KN*; the charge states of impurities are expressed in units of the elementary charge *e*. It is assumed that the impurities are randomly (Poissonian) distributed over the crystal, and the electrical neutrality condition is satisfied, *N*_{+1} = *KN*, where 0 ≤ *K* ≤ 1 is the compensation ratio.

In the phenomenological theory of the molecular field, the spin magnetic moment of an electron localized on a hydrogen-like impurity atom experiences an orienting effect not only from the external magnetic field with the induction *B*_{0} = *μ*_{0}*H*_{0}, but also from, induced by the nearest spins, an effective “exchange” magnetic field with the strength

where *μ*_{0} is the magnetic constant, *λ* is the coefficient of proportionality, and *M* is the total magnetic moment.

The intensity of the ESR signal of donors is proportional to the difference in the concentration of donors with the spin magnetic moments of electrons in the direction (0↑) and opposite to the direction (0↓) of the external magnetic field induction, i.e., *N*_{0↑} − *N*_{0↓}.

On the one hand, the total electron spin magnetic moment of all donors in the charge state (0) with spin *S* is given by the following formula:^{13}

where *N*_{0} = *N*_{0↑} + *N*_{0↓} = (1 − *K*)*N* is the concentration of donors in the charge state (0) at low temperatures, *g*_{d} ≈ 1.57 is the *g*-factor of an electron of the hydrogen-like donor impurity of arsenic in the germanium crystal, which is not involved in the covalent chemical bonding,^{3} *B*_{0} = 425 mT is the induction of the external constant magnetic field used in ESR experiments in Refs. 3–5, *k*_{B} is the Boltzmann constant, *T* is the absolute temperature, *k*_{B}*T* is the thermal energy, *μ*_{B} is the Bohr magneton, and *B*_{S}(*x*) is the Brillouin function; *B*_{1/2}(*x*) = tanh *x* for *S* = 1/2.

On the other hand, the total electron spin magnetic moment of donors in the charge state (0) is determined by the number of uncompensated spin moments (see, e.g., Refs. 14 and 15),

where *g*_{d}*μ*_{B}/2 is the modulus of the magnetic moment of the donor electron in the charge state (0).

From Eq. (4), taking into account *N*_{0↑} + *N*_{0↓} = *N*_{0}, we obtain the concentration of electrically neutral donors with the electron spin directed along (0↑) and against (0↓) the field

To take into account the effective exchange magnetic field *B*_{m} acting on the electron spins of donors, in Eq. (4), we should write *B*_{0} + *B*_{m} instead of *B*_{0}, so that

For weak magnetic fields (*g*_{d}*μ*_{B}*B*_{0} ≪ *k*_{B}*T*), confining ourselves to the first term in the expansion of tanh *x* in a series at *x* ≪ 1 and passing, using the relations *B*_{0} = *μ*_{0}*H*_{0} and *B*_{m} = *μ*_{0}*H*_{m}, to the magnetic field strengths *H*_{0} and *H*_{m}, from Eq. (6), we obtain

Then, according to Eq. (3), taking into account Eqs. (1) and (7), the total magnetic moment of electrically neutral donors with uncompensated electron spin magnetic moments is

From Eq. (8), it follows that

Then, taking into account Eq. (9), the dimensionless paramagnetic susceptibility of electrically neutral donors is

Here, the Curie constant *C* and the Curie–Weiss temperature Θ are given by the following relations:

where Θ > 0, Θ = 0, and Θ < 0 correspond to the ferromagnetism, paramagnetism, and antiferromagnetism of the system of electron spins of donors in the charge state (0), respectively.

Note that the dimensionless quantity *χ*_{d} according to Eq. (10) is written in SI units. The relationship between the value $[\chi d]CGS$, written in the CGS (Gaussian) system of units, and the value $[\chi d]SI$, written in the SI system, has the form:^{16} $[\chi d]SI=4\pi [\chi d]CGS$.

The purpose of this work is to describe the Curie–Weiss low-temperature behavior of the impurity paramagnetic susceptibility *χ*_{d} in diamagnetic covalent semiconductors, doped with donors up to the insulator–metal phase transition, and to study the effect of the doping level and the degree of compensation on this behavior. First, we have found the relation between the strength of the effective internal magnetic field *H*_{m} and the total magnetic moment *M* of neutral donors. Then, the Curie–Weiss temperature Θ is calculated and its behavior is studied depending on the doping level and the degree of compensation. In conclusion, we discuss main results using the predictions of the proposed model to be compared with the experimental data on the *n*-Ge:As doped up to the metal–insulator phase transition and compensated by Ga acceptor impurities.

## II. EFFECTIVE INTERNAL MAGNETIC FIELD STRENGTH

According to Ref. 11, the magnitude of the effective internal magnetic field of strength **H**_{m} is found in a self-consistent way from the condition of the equality of the Heisenberg Hamiltonian $HH$, containing this field inside the exchange integral, and the Hamiltonian of the Zeeman splitting $HZ$ under this field. In our case, they can be represented as

where *J* is the exchange integral between a pair of the nearest-neighbor (first nearest) donors in the charge state (0), **S**_{i} is the spin moment of the donor in the charge state (0) with spin *S*_{i} (in units of the Planck constant *ℏ*), *z* is the number of the nearest neighbors (electrically neutral donors with an average spin moment ⟨**S**_{j}⟩) for a given neutral donor with spin *S*_{i}.

Setting $HH=HZ$, according to Eq. (12), we find

Since the total magnetic moment is **M** = *N*_{0}*g*_{d}*μ*_{B}⟨**S**_{j}⟩, and the strength of the internal magnetic field is **H**_{m} = *λ***M**, from Eq. (13), we obtain

Note that to calculate the value of *J* for the electron spins (quasi)localized on donors, one cannot use the formulas for the exchange energy of *c*-band electrons according to the model in Ref. 17. Moreover, within the framework of the Ising model, a numerical simulation of the direct exchange interaction of magnetic impurities (e.g., Mn) in nonmagnetic semiconductors was carried out recently in Ref. 18. It was shown in this work that at low temperatures, regardless of the sign of the value of the interimpurity exchange interaction (i.e., the sign of the *J* integral), the Curie law for the impurity magnetic susceptibility is not satisfied. The issue arises due to the formation of a spin glass phase by the uncompensated electron spins of impurities.

Substituting Eq. (14) for the proportionality coefficient *λ* (between the spin magnetic moment **M** of a set of electrically neutral donors and the strength of the internal magnetic field **H**_{m}) into Eq. (11), we obtain the Curie–Weiss temperature in the following form:

where *z* = 1 is the number of donors in the charge state (0) nearest to the given donor, which is also in the charge state (0); the exchange integral *J* is calculated in Sec. IV.

## III. HEITLER–LONDON MODEL FOR A PAIR OF ELECTRICALLY NEUTRAL DONORS

According to the Heitler–London model (see, e.g., Ref. 19), the solution of the Schrödinger equation for the H_{2} molecule gives two energy levels (singlet and triplet) with the values specified only by the ionization energy of the hydrogen atom and the distance between two protons in this electroneutral molecule. Let us apply this approach to a system of two electrically neutral donors (and two electrons localized on them).

The singlet state of the system is characterized by symmetric spatial and antisymmetric spin wave functions (the product of a spatial wave function and a spin wave function should be antisymmetric due to the Pauli principle). The energy of the molecular system “two donors in the charge state (0)” in a singlet state is^{20,21}

where

where $Ed\u0304$ is the average thermal ionization energy of a donor, which depends on the concentration of donors and acceptors; *ρ* = *r*/*a*_{H} is the ratio of the distance *r* between the centers of two electrically neutral donors to the Bohr radius $aH=e2/8\pi \epsilon r\epsilon 0Id$ for a single donor with the ionization energy *I*_{d}; *e* is the elementary charge; *ɛ*_{r} is the relative dielectric constant of the crystal matrix; *ɛ*_{0} is the electric constant; *γ* = 0.577 22 is the Euler constant; *M*_{d} = (1 − *ρ* + *ρ*^{2}/3)exp(*ρ*); $Ei(x)=\u2212\u222b\u2212x\u221et\u22121\u2061exp(\u2212t)dt$ is an exponential integral function.^{22}

The triplet state of the system “two donors in the charge state (0)” is described by the antisymmetric spatial and symmetric spin wave functions with the energy^{20,21}

The difference in energies of the triplet (t) and singlet (s) states of electrons on a solitary pair of donors [both in the charge state (0)] according to Eqs. (16) and (17) is

The effective width of the donor band^{23} for a Gaussian distribution of donor energy levels relative to the energy level *I*_{d} of a single donor can be written as^{24}

where 2*KN* is the total concentration of donor (*N*_{+1}) and acceptor (*KN*) ions at low temperatures (when *N*_{+1} = *KN*). When calculating *W*_{d}, only the Coulomb interaction of the nearest impurity ions randomly (Poissonian) distributed in the crystal was taken into account.

Note that, according to Refs. 24 and 25, at low temperatures, the concentration of *c*-band electrons is *n* ≪ *K*(1 − *K*)*N* and the root-mean-square fluctuation of their potential energy *W*_{n} ≪ *W*_{d}. The inequality *W*_{n} ≪ *W*_{d} is due to the fact that the “cold” electron is not affected by potential energy fluctuations on spatial scales smaller than its average wavelength (according to de Broglie), which is much greater than the average distance between impurity ions.

For a broad, in comparison with the thermal energy, donor band (*W*_{d} ≫ *k*_{B}*T*), the average thermal ionization energy of the donor in the charge state (0) is determined by the following expression (see Ref. 26 and also Refs. 25 and 27 for an acceptor in a *p*-type crystal):

where $Id=e2/8\pi \epsilon r\epsilon 0aH=14.17$ meV is the thermal ionization energy of a single hydrogen-like donor (an As atom replacing a Ge atom in the crystal structure),^{28} Δ_{d} is the shift of the Fermi level relative to the energy level *I*_{d} of a single donor, and *δE*_{c} < 0 is the shift of the bottom of the *c*-band to the bandgap (further calculated in two ways).^{29}

For low temperatures, the quantity Δ_{d} is found from the equation of electrical neutrality of the covalent crystal: *N*_{+1} = *KN*, where *N*_{+1} = *N* − *N*_{0} is the volume concentration of ionized donors and *K* is the ratio of the compensating impurity concentration to the main (dopant) impurity concentration. Thus, taking the Gaussian distribution of donor energy levels with respect to *I*_{d}, for *W*_{d} ≫ *k*_{B}*T*, we obtain^{27,30}

where erf(·) is the error function;^{22} *W*_{d} > 0 is determined by Eq. (19); the quantity Δ_{d} < 0 for *K* < 0.5, Δ_{d} = 0 for *K* = 0.5, and Δ_{d} > 0 for *K* > 0.5.

Note that the quantity Δ_{d}, according to Eq. (21), is determined by the concentration of donors *N* and the compensation ratio *K*. As formally follows from Eq. (21), for the limiting cases of the compensation ratios *K* → 0 and *K* → 1, the ratio Δ_{d}/*W*_{d} → −*∞* and +*∞*, respectively.

The shift of the bottom of the *c*-band *δE*_{c} < 0 to the bandgap is calculated in the quasi-classical approximation of quantum mechanics accounting for two possible factors.

- Restriction of the region of electron localization on the donor,
^{31}where(22)$\delta Ec1=\u2212IdaHRim<0,$*R*_{im}= 0.62[(1 +*K*)*N*]^{−1/3}is the average radius of a spherical region per impurity atom in a crystal matrix.Formula (22) gives the value of the lowering of the bottom of the

*c*-band to the bandgap (*δE*_{c1}< 0) due to collectivization of the excited states of donors in the charge state (0) and their merging with the*c*-band states with the donor concentration and the compensation ratio (due to decreasing*R*_{im}). - Screening (in the Debye–Hückel approximation) of the Coulomb field of an ionized donor by electrons hopping between donors in the charge states (0) and (+1) against the background of stationary negatively charged acceptor ions,
^{30,32}where Λ(23)$\delta Ec2=\u2212Id3aH2(\Lambda sc+Rim)<0,$_{sc}is the screening length (radius) of the impurity ion. Formula (23) gives a decrease in the energy of the affinity of the

*c*-band electron to the donor in the charge state (+1). The value of*δE*_{c2}is equal to the electrostatic energy of the electrically neutral system “donor ion + cloud of locally uncompensated negative charges screening it.” The charge states of donors seem to migrate over the crystal due to hopping of electrons with the effective concentration^{33}*N*_{0}*N*_{+1}/*N*= (1 −*K*)*KN*. In this case, acceptors are in the charge state (−1) and are immobile.

At low temperatures (when the donor bandwidth *W*_{d} is much greater than the thermal energy *k*_{B}*T*), the screening length Λ_{sc} is described by the following expression:^{27,30}

where the quantity Δ_{d}/*W*_{d}, for the known compensation ratio *K* of donors by acceptors, is found from Eq. (21).

From Eq. (24), taking into account Eq. (21), it follows that Λ_{sc} → *∞* in the limiting cases *K* → 0 and *K* → 1, when the hopping migration of electrons between donors in the charge states (0) and (+1) is impossible (there are no free places or no electrons on donors, respectively). In this case, the effective concentration of electrons hopping between donors (1 − *K*)*KN* → 0. For *K* = 0.5, when Δ_{d} = 0, the screening length is minimal: Λ_{sc} = 0.725*N*^{−1/3}.

By comparing Eqs. (22) and (23), we conclude that the shift of the bottom of the *c*-band |*δE*_{c1}| > 0 due to the restriction of the region of electron localization on each donor in the charge state (0) is greater than the shift of the bottom of the *c*-band |*δE*_{c2}| > 0 due to the screening of the Coulomb field of each donor in the charge state (+1).

Let us estimate the applicability of Eq. (20) taking into account Eqs. (21)–(23). So, according to Eq. (20), the value of $Ed\u0304$ goes to zero for *K* = 0.5 ($\Delta d$ = 0) and *δE*_{c} = *δE*_{c1} by Eq. (22) at the concentration of donors (arsenic atoms) *N*_{1} ≈ 4.4 · 10^{18} cm^{−3}, while for *δE*_{c} = *δE*_{c2} by Eq. (23)—at the donor concentration *N*_{2} ≈ 1.7 · 10^{19} cm^{−3}. Let us now compare the values of the concentrations *N*_{1} and *N*_{2} with the critical concentration of arsenic *N* = *N*_{M}, corresponding to the transition of a germanium crystal from the insulator state to the metallic one (the Mott transition).

According to Ref. 34, the quantity *N*_{M} is determined by the relation

where *ɛ*_{r} = 15.4 is the low-temperature low-frequency relative permittivity of the germanium crystal.^{35}

Calculations using Eq. (25) are in general quantitatively consistent^{34} with the experimental data for *n*- and *p*-type silicon and germanium crystals doped with hydrogen-like impurities over a wide range of dopant concentrations and compensation ratios. As follows from Eq. (25), for *n*-Ge:As,Ga at *K* = 0.5, the calculated critical concentration of arsenic is *N*_{M} ≈ 5.1 · 10^{17} cm^{−3}. [The experimental value is *N*_{M} ≈ 3.7 · 10^{17} cm^{−3} for *K* → 0 (see Ref. 34 and references therein).] It is evident that the values *N*_{1} and *N*_{2} obtained from the condition $Ed\u0304=0$ for *δE*_{c} = *δE*_{c1} by Eq. (22) and for *δE*_{c} = *δE*_{c2} by Eq. (23) considerably exceed the value of the critical concentration *N*_{M} from Eq. (25). In other words, the insulator–metal concentration phase transition occurs in the donor band, before it merges with the bottom of the *c*-band. This was experimentally shown for the first time in Ref. 36, which played an important role in the formation of ideas about the Mott transition upon doping and compensation of semiconductors (see also Ref. 37).

## IV. CALCULATIONS OF EXCHANGE INTEGRAL AND CURIE–WEISS TEMPERATURE

The probability that an impurity atom randomly selected among all the impurity atoms is a donor in the charge state (0) is (1 − *K*)/(1 + *K*). The fractions of neutral donors having the magnetic moment oriented along and against the direction of the external magnetic field are *N*_{0↑}/*N*_{0} and *N*_{0↓}/*N*_{0}, respectively. We assume that donors in the charge states (0) and (+1), as well as acceptors in the charge state (−1), are randomly distributed over the crystal (specifically, with the Poisson distribution; see, e.g., Refs. 24 and 38). In this approximation, the probability $P0,0(\rho )d\rho $ that two donors are the nearest-neighbor electrically neutral donors is the product of three probabilities: (i) the probability $4\pi aH3\rho 2(1\u2212K)Nd\rho $ to find a donor in the charge state (0) in a spherical layer with radius *r* = *a*_{H}*ρ* and thickness d*r* = *a*_{H} d*ρ*, (ii) the probability (1 − *K*)/(1 + *K*) that an impurity atom in the center of the sphere is a donor in the charge state (0), and (iii) the probability $exp[\u2212(4\pi /3)(aH\rho )3(1\u2212K)N]$ that there are no other donors in the charge state (0) in a ball of radius *r*, that is,

The exchange integral *J* in Eq. (15) is found as the average exchange energy per two nearest (the first nearest) electrically neutral donors in a germanium crystal,

where $N02=(N0\u2191+N0\u2193)2=N0\u21912+2N0\u2191N0\u2193+N0\u21932$; the quantities *N*_{0↑} and *N*_{0↓} are determined according to Eq. (5); $Et\u0304$ and $Es\u0304$ are the average energies of the triplet (t) by Eq. (17) and singlet (s) by Eq. (16) states of electrons on a pair of donors in the charge state (0). The quantities $Et\u0304$ and $Es\u0304$ are calculated as follows:

where $P0,0(\rho )$ is given by Eq. (26).

Note that in Eq. (27), the ratios $(N0\u2191/N0)2$ and $(N0\u2193/N0)2$ are the probabilities that both of the nearest donors in the charge state (0) have spins co-directed and oppositely directed to the induction of an external field, respectively (i.e., they are in the triplet state), and $2N0\u2191N0\u2193/N02$ is the probability that the electron spins in a pair of the nearest donors in the charge state (0) are oppositely directed (i.e., they are in a singlet state). The values of *E*_{s}(*ρ*) and *E*_{t}(*ρ*) are calculated by Eqs. (16) and (17), respectively.

The signs of the terms inside the square bracket in Eq. (27) are chosen in such a way that in the triplet state the energy of the system of two electrons is higher than in the singlet state.^{39}

We consider an *n*-type germanium doped with arsenic atoms and compensated with gallium atoms on the insulator side close to the Mott transition. Let us estimate the characteristic temperature *T*_{m}, below which it is possible to observe the magnetic ordering of donors in the charge state (0) with uncompensated electron spin magnetic moments. We assume that for the manifestation of spin ordering, it is necessary that the average acoustic phonon energy $Eac\u0304$ does not exceed the average splitting between the triplet and singlet states |$Et\u0304\u2212Es\u0304$| of a pair of the nearest donors in the charge state (0). It can be shown (following, e.g., Refs. 40 and 41) that at temperatures much lower than the Debye temperature in germanium crystals (*T*_{D} = 374 K; see Ref. 28), the average acoustic phonon energy in the Debye approximation equals $Eac\u0304=(\pi 4/36)kBTm$.

Then, taking into account $Eac\u0304\u2264|Et\u0304\u2212Es\u0304|$ and Eq. (28), the temperature *T*_{m} is determined by the following relation:

where $P0,0(\rho )$ is given by Eq. (26); the quantity *δE*_{0,0}(*ρ*) is determined by Eq. (18). For *T* ≪ *T*_{D}, the concentration of acoustic phonons is $Nac\u224821.6NGe(T/TD)3$, where *N*_{Ge} ≈ 4.4 · 10^{22} cm^{−3} is the concentration of intrinsic atoms in a germanium crystal. For *T* = 4.2 K, we have *N*_{ac} ≈ 1.35 · 10^{18} cm^{−3}, which slightly exceeds the critical concentration of donors in germanium for the insulator–metal transition [see Refs. 3 and 4 and Eq. (25)].

Figure 1 shows the calculated Curie–Weiss temperature Θ and specific paramagnetic susceptibility *χ*_{d}/*d*_{Ge} of *n*-Ge:As,Ga crystals as a function of the dopant (arsenic) concentration *N* for the compensation ratio *K* = 0.5 (which turns out to be optimal for the manifestation of ferromagnetic spin ordering) and the temperature *T* = 3 K; the density of germanium is *d*_{Ge} = 5.3234 g/cm^{3} (see Ref. 28). It can be seen that at doping levels considerably lower than the critical values for the insulator–metal transition, the Curie–Weiss temperature falls to the region of ultralow temperatures below 1 K, while the deviations from the Curie law are insignificant. This creates particular difficulties during experimental studies of the phenomena associated with electron spins.

Figure 2 shows the calculated Curie–Weiss temperature Θ and specific paramagnetic susceptibility *χ*_{d}/*d*_{Ge} as a function of the compensation ratio *K* for the concentration of arsenic atoms *N* = 3 · 10^{17} cm^{−3} and *T* = 3 K. It can be seen that the model predicts sign changing of the Curie–Weiss temperature from minus to plus (low-temperature transition from antiferromagnetic to ferromagnetic local ordering of impurity spins) in the region of 15%–30% compensations, reaching (close to half compensations) the maximum positive values Θ ≈ 1.3 K with the following decrease (transition to paramagnetism) in the region of high compensations, when the spin exchange is weakened. The change in the sign of the Curie–Weiss temperature is associated with the change in the sign of the difference between the average energies of triplet and singlet states $Et\u0304$ and $Es\u0304$ [Fig. 2(a)] at relatively small compensation ratios of the order of ten percent.

## V. COMPARISON WITH EXPERIMENT FOR COMPENSATED GERMANIUM

Figure 3 shows (in SI units) the temperature dependences for the specific paramagnetic susceptibility *χ*_{d}/*d*_{Ge} of compensated *n*-Ge crystals, where *d*_{Ge} = 5.3234 g/cm^{3}. Points represent the ESR measurements from Ref. 5 at *B*_{0} = 425 mT of the magnetic susceptibility of *n*-Ge:As,Ga samples with the arsenic concentration *N* = (3.97–4.49) · 10^{17} cm^{−3}, compensated by gallium atoms with the compensation ratio *K* = 0.28–0.58. Solid lines are calculations for *N* = 4.33 · 10^{17} cm^{−3} and *K* = 0.49 according to Eq. (10) taking into account *δE*_{c} by Eq. (22) (curve *1*) and *δE*_{c} by Eq. (23) (curve *2*). Dashed line *3* indicates a calculation of the paramagnetic susceptibility according to Eq. (10) at Θ = 0 K (Curie law) for the same values of *N* and *K*.

Note that according to Eq. (19), for the donor concentration *N* = 4.33 · 10^{17} cm^{−3} and the compensation ratio *K* = 0.49, the impurity bandwidth *W*_{d} is equal to 18.5 meV. Thus, for *T* = 3 K, we have *W*_{d}/*k*_{B}*T* ≈ 72, i.e., relation (21) is valid.

As seen, the Curie–Weiss behavior of the paramagnetic susceptibility predicted by the proposed model has a fairly good quantitative agreement with the experimental data from Refs. 2–5 for the samples of compensated *n*-Ge:As,Ga close to the insulator–metal phase transition. Note an agreement of the functional dependence on the compensation ratio, for which the proposed model predicts maximal values of *χ*_{d} at the compensation ratios ≈40%–50% (for *N* = 3 · 10^{17} cm^{−3}).

Thus, the constructed model provides the behavior of specific paramagnetic susceptibility *χ*_{d}/*d*_{Ge} that corresponds to the transition from the Curie law to the Curie–Weiss law (Curie–Weiss behavior) at temperatures below *T*_{m}.

The experiment (see Ref. 5) shows that the increase in paramagnetic susceptibility observed with a decrease in temperature from 5 to 3 K at lower temperatures ceases and it is replaced by the Curie law for electrons localized in the donor band, and at higher temperatures it transforms into a temperature-independent plateau (Pauli paramagnetism). Such a behavior is caused by the formation of a narrow Coulomb gap in the compensated samples on the insulator side of the insulator–metal phase transition. At temperatures higher than the Coulomb gap width, the gap does not manifest itself in the spin and transport properties. At even higher temperatures, when the energy distribution of electrons covers also the *c*-band states (electrons are not degenerate), the Curie law is observed again.

## VI. CONCLUSION

The antiferromagnet–ferromagnet–paramagnet transition of a diamagnetic semiconductor doped with paramagnetic hydrogen-like donors is described theoretically for the first time as a function of the temperature *T* and the compensation ratio *K*. (For an *n*-type semiconductor, the quantity *K* is the ratio of the concentration of compensating acceptors *KN* to the concentration of doping donors *N*.) The Curie–Weiss behavior of the low-temperature paramagnetic susceptibility of compensated semiconductors on the insulator side of the insulator–metal phase transition is shown. For the analytical specification of the density of states in the impurity band, we used model concepts developed so far only for moderately doped semiconductors.

Calculations of the energy of exchange interaction of two electrons localized on the nearest two neutral donors are based on the Heitler–London model. Additionally, the shift of the bottom of the *c*-band to the bandgap is taken into account due to the separate influence of two factors: (i) due to confinement (limitation of the localization region) of the electron on the donor and (ii) due to screening of the Coulomb field of the donor ion by electrons hopping between hydrogen-like donors.

The model shows that the magnetic susceptibility satisfies the Curie–Weiss law with the Curie–Weiss temperature, which grows with compensation, changes sign from minus to plus in the compensation region of 15%–30%, and then reaches values on the order of several kelvin at compensations of about 50% for doping levels in Ge close to the insulator–metal phase transition.

The applicability of the model to the experimental results obtained using the ESR spectroscopy technique requires a small Zeeman splitting of donor energy levels in the magnetic field with induction *B*_{0} = 425 mT in comparison with the thermal energy.

The behavior of the paramagnetic susceptibility predicted by the model at low temperatures (*T* ≈ 2–7 K) is in good agreement with the results of measurements performed using the ESR technique on a series of samples of compensated *n*-Ge:As,Ga close to the insulator–metal phase transition.

However, considering the experimental data on the magnetic susceptibility behavior in a wider temperature range (see Ref. 5), a qualitative difference is also seen. In experiments, for the temperatures *T* > 7 K, insulating (at lower temperatures) samples show a distinct transition to the Pauli paramagnetism, which is characteristic for “metalized electronic states” of the donor band. This is associated with the disappearance of the manifestation of a narrow Coulomb gap at the Fermi level in the donor band when the thermal energy is comparable to the width of this band. More precisely, at higher temperatures, when electrons pass from the states of the donor band to the *c*-band, the Curie paramagnetism gives way to the Pauli paramagnetism. The proposed model is limited to considerably lower temperatures at which both of the thermal effects indicated here do not occur.

## ACKNOWLEDGMENTS

This work was supported by the Belarusian National Research Program “Materials Science, New Materials and Technologies” and by the Belarusian Republican Foundation for Fundamental Research (Grant No. F21RM-137).

The authors declare no competing financial interest.

## DATA AVAILABILITY

The data that support the findings of this study are openly available in Mendeley Data at http://doi.org/10.17632/vmkywd69cr, Ref. 42.

## REFERENCES

At low temperatures, the average kinetic energy of electrons in the *c*-band is much less than the thermal ionization energy of the impurity, and they turn out to be “frozen out” to the states of the donor band.

We consider the case of the classical broadening of the donor band, when the spread of the donor energy levels due to the Coulomb interaction of the nearest impurity ions is much greater than the quantum broadening of their energy levels due to the finite time of electron localization on the donor.

The change in the thermal ionization energy of a donor (As) in germanium crystals under the magnetic field with induction *B*_{0} = 425 mT can be neglected.