Effect of thermodynamic fluctuations of magnetization on the bound magnetic polaron state in ferromagnetic semiconductors

We extend the theory of the bound magnetic polaron (BMP) in diluted paramagnetic semiconductors to the situation with a ferromagnetic phase transition. This is achieved by including the classical Gaussian fluctuations of magnetization from the quartic (non-Gaussian) term in the effective Ginzburg–Landau Hamiltonian for the spins. Within this approach, we find a ferromagnetically ordered state within the BMP in the temperature range well above the Curie temperature for the host magnetic semiconductor. Numerical results are compared directly with the recently available experimental data for the ferromagnetic semiconductor GdN. The agreement is excellent, given the simplicity of our model, and is because the polaron size ( ≃ 1.4 ?> nm) encompasses a relatively large but finite number ( N ≈ 400 ?> ) of quasiclassical spins S = 7 / 2 ?> coming from Gd3+ ions. The presence of BMP invalidates the notion of critical temperature and thus makes the incorporation of classical Gaussian fluctuations sufficient to realistically describe the situation.


Introduction
The properties of a shallow donor electron or acceptor hole in the presence of localized d 3 or f 4 electrons (termed throughout the paper as the spins) have been studied intensively experimentally [1][2][3][4][5][6][7] and theoretically [7][8][9][10][11][12][13][14][15][16][17][18] during the last four decades. The resultant bound state, for which both the electrostatic and the magnetic energy contributions are important, is called a bound magnetic polaron (BMP). These studies elucidated the role of the − s d ( − s f ) exchange coupling of individual carriers to the non-uniformly distributed localized magnetic moments. More recently, the important role of these quantum systems has been discussed for materials encompassing diluted magnetic semiconductors, [19][20][21][22][23], ferromagnetic perovskites [24], and ferromagnetic semiconductors [25,26]. BMPs, being objects of nanoscale size, have attracted a growing interest in the studies of various quantum structures [27][28][29][30][31][32][33]. The origin of ferromagnetism in DMSs [34][35][36][37][38] represents a basic question about the dominant role of antiferromagnetic superexchange in all of those materials when the carrier concentration is zero or small. Within a semiclassical approximation to the spins, one can consider even the limiting case of magnetic ions concentration x = 1 and thus encompass the limit of a ferromagnetic semiconductor. This, in turn, allows us to address the question of the validity of results for DMSs concerning BMP states also in the ferromagnetic phase. In this manner, one can question the validity of the results coming from the mean-field theoretical treatment developed by Kuivalainen et al [10], which have shown the appearance of the spin polarization within the BMP orbit at temperatures well above the critical temperature T c of the host ferromagnetic semiconductor. The same effect, though not in such a clear form, has been observed in La −x 1 Sr x MnO 3 [24]. One should also mention precursory theoretical studies [39] on autolocalization of the conduction band electrons due to magnetization fluctuations.
The purpose of this work is to address these important issues. To do this, we extend essentially a single BMP theory applicable to a paramagnetic DMS, developed by Dietl and Spałek (DS) [11] and others [12][13][14][15][16][17], to be applicable to a magnetic semiconductor with the ferromagnetic phase-transition (Curie) point. This is achieved by including the quartic term in magnetization in the system free energy functional for the spins. Within this approach, incorporating the effective Gaussian fluctuations around the mean-field ordered state, we also find a magnetically ordered ground state of the BMP in a temperature range well above the Curie-Weiss temperature of the host magnetic semiconductor. Our findings are quantitatively consistent with the experimentally observed magnetization [25,26], as well as with previous theoretical investigations [10,39]. Numerical calculations are presented for the values of material parameters corresponding to GdN and compared directly to recent experimental data [26].
The structure of the paper is as follows. In section 2 we formulate the theoretical approach with inclusion of fluctuations of magnetization due to the localized spins. In section 3 we compare the numerical analysis with experimental data in a quantitative manner. In section 4 we overview the entire approach. In appendix A, we provide a brief overview of the derivation of the effective Ginzburg-Landau (G-L) functional for the spins and estimate its parameters.

Effective Hamiltonian with fluctuating variables
We start by considering a ferromagnetic crystal consisting of equal-magnitude localized magnetic moments distributed over lattice sites i { }, each possessing spin = S S.
i Magnetic properties of the host spins are described by the Heisenberg Hamiltonian and, for the sake of simplicity, we limit ourselves to the magnetic ions with quenched orbital moment, L = 0. The system also contains isolated shallow donors with electronic states that are described with the help of the effective mass approximation. With this formulation, the model has quite a wide range of applications, e.g., BMPs in ferromagnetic semiconductors, insulators, or DMSs, provided that the distance between the donors is substantially larger than their hydrogenic-like orbital size. Furthermore, within the continuous-medium approximation for the spins, one can write the donor electron Hamiltonian in the form [11] where  1 is the single-particle part of the Hamiltonian and the polaron effective spin splitting Δ φ M [ ; ] is defined as follows:

B a c l
where g * is the Landė factor for the donor electron, H a is the applied magnetic field, σ denotes the Pauli matrices, and J c is the exchange integral of the contact Fermi (s-d(f )) interaction between localized spins  S { } i and those of impurity carrier ŝ l . The spin density is defined as , with the summation running over the sites occupied by the magnetic ions.
The assumptions that the exchange field is weak and that orbital moments are absent (as for e.g., Mn 2+ or Eu 2+ ions) allow us to write the electron wave function as a product of functions that depend on the spatial and the spin variables separately: is the wave function of spins and χ σ is the corresponding spin part for the carrier, whose spatial wave function is φ r ( ). Using such wave function, one can average equation B 0 is the local magnetization per unit volume; = N n v 0 0 0 is number of atoms per unit volume containing fraction x of magnetic atoms, α ≡ J v n c 0 0 is the effective exchange constant, and γ σ is the polarization of the carrier spin. Consequently, we average now equation (4) and redefine  e in the following manner: is the resultant splitting of the donor state, with B a B 0 0 playing the role of the homogeneous part of the spin splitting for the electron [11], whereas M 0 is the static part of the magnetization, i.e. the magnetization corresponding to the cation volume due to the average spins polarization. Note, that direction of Δ 0 defines a convenient quantization axis due to colinearity of H a and M 0 . The procedure of averaging equation (5), with respect to the fluctuating part of r M( ), is as follows. We introduce the G-L functional for the spins, here taken with the quartic term where μ ξ , , and λ are the model parameters of the functional (see appendix A for its derivation from the Heisenberg model), μ is the inverse static magnetic susceptibility, ξ determines the characteristic (correlation) length, and the term λ ∼ introduces a quartic contribution to the system properties. Because we would like to analyze the role of thermodynamic fluctuations of magnetization on the BMP state in the vicinity of the magnetic phase transition, we consider both the longitudinal η ∥ r ( ) and the transverse, The corresponding contributions from r M ( ) 2 are as follows: and to the quartic term In other words, we consider the quartic contribution to r M( ), but retain only the quadratic (Gaussian) fluctuations around this (non-zero) static value. The feature will lead to a spontaneous magnetic overall polarization within the polaron volume, even well above bulk T c . Therefore, even in an isotropic solid, the appearance of ≠ M 0 0 breaks the spin rotational symmetry and introduces a spontaneous polarization of the system. Substituting these expressions into  S give us the following explicit quadratic contributions:  where the second and the third lines define  S and the contribution from  M · 0 to  M [ ] S 0 is zero. In effect, we have a full G-L functional to the fourth-order (the first term) and the Gaussian fluctuation contributions around an essentially mean-field state.

Spin splitting distribution
Next, we explore the use of the important simplification by assuming that Thus, we effectively account for only the quadratic (Gaussian) contribution to the magnetization fluctuations appearing around the resultant static value. Within this approximation, the following relation holds: In other words, the contribution λ ∼ in the fluctuation part has been absorbed into the Gaussian part. This amounts to saying that the thermodynamics related to  η μ λ [ ; , ] S can be determined directly from the well-known solution of the anisotropic Gaussian model, as carried out in [11]. To obtain the renormalized by fluctuations Hamiltonian for the donor electron, we follow that reference and define next the probability distribution of the magnetization fluctuations within electron in state φ r ( ): where η r D ( ) symbolizes the functional integration over all possible space profiles of η r ( ). Then the probability distribution of electron energies P E ( ) R is given by . Note that we may use the preceding equation to define the effective renormalized Hamiltonian of the BMP in the following manner: The probability distribution of electron energies P E ( ) R is difficult to calculate. A significantly simpler task is to calculate the probability distribution Δ P ( ) of the spin splitting Δ, which is obtained by the appropriate change of variables, namely The meaning of the transformation is explained in [11] and its execution provides ⎛ To calculate the partition function Δ Z [ ], one considers the fact that within our approximation the presence of equilibrium magnetization M 0 influences Δ Z [ ] by modifying the corresponding contribution μ μ → . Finally, performing analogous calculations as in [11], the partition function Δ Z [ ] can be expressed as follows: with the propagator ′ r r Q ( , ) i explicitly given by the Ornstein-Zernike correlation function In expression (20), important parameters of the present approach appear, namely ϵ pi , with i = 1, 2, and 3. These quantities, for the case with λ = 0, reduce to ϵ p of DS theory. This is not surprising because the DS theory may be regarded as the Gaussian limiting case (λ = 0) of the present approach. Note also, that it may be energetically favorable for the BMP to create a spontaneous polarization ≠ M 0 0 of the spins only within its effective volume πa 3 when λ ≠ 0. We discuss this feature of our results in the next section.

Thermodynamic properties
The next step, in analogy to [11], is to define a contribution ΔF to the free energy of the system, introduced by the presence of the donor electron that had formed the BMP state. Here, ΔF is defined as the difference between the free energy when donor is present and the spin part when donor is absent, i.e., ln Tr exp exp .  where M 0 is the static magnetization density and n 0 is the BMPs concentration. In the preceding expression, the second term represents the magnetic free energy density of spins considered here in the simple Landau approximation. Above T , c it represents demagnetization energy. It should be noted that, that in [39], an expression for ΔF using the Feynman path integral method has been used to consider the spin polaron formation. Therefore, considering a BMP as a trapped conduction electron on the hydrogenic-like center, one can look to [39] as a precursor work. It should be stressed that the dependence of ΔF on M 0 has not been studied thus far.
Before presenting numerical results, we consider here a simple case with the parameter ξ → 0. In such a case, an analytical expression for ϵ pi can be determined. Namely, we have: pi i 2 where the constant α is defined by the following expression: Finally, the partition function Δ Z [ ] corresponding to ΔF may be written as follows: where θ is the angle between Δ and Δ . 0 From equation (27), one can see that ϵ pi is inversely proportional to dressed μ , i which in turn includes the contribution proportional to λM 0 2 . Therefore, the quadratic contributions to the magnetization fluctuations, originating from the quartic-term contribution in the effective functional for spins, are important.

Numerical results and comparison with experiment
We apply our model to the recently published experimental data on epitaxial films of GdN with varying doping levels [26]. In this ferromagnetic semiconductor, Gd 3+ ions have spin = S 7 2 due to the half-filled f 4 shell, with a net magnetic moment of μ 7 B per Gd 3+ ion. The carrier states originate from either hybridized 5d-6s states due to Gd or shallow donor states created on the charged nitrogen vacancies. Experimentally, the Curie temperature = T 37 c K for the undoped semiconductor [26] increases with increasing doping, e.g., for a sample with carrier concentration at room temperature = × − n 3.3 10 20 cm − , 3 we see a sharp increase to the value of = T 70 c K. The characteristic feature of this magnetic-semiconductor thin epitaxial film is the double dome structure of various physical quantities plotted vs. temperature, which has been ascribed to the presence of BMPs [26].
First, we determine the values of material parameters entering the G-L functional for the host magnetic semiconductor, namely μ ξ , , and λ. The inverse magnetic susceptibility can be expanded in the power series of λ, namely [40] ∫ where Λ is the integration cutoff. For the case with ξ = 0 that we consider here, the integral is proportional to the volume of the reciprocal space, and thus is regarded as negligible. Therefore, in such a situation μ β χ = . (31) Next, we assume that the magnetic susceptibility obeys the Curie-Weiss law: M c with the molar Curie constant given by In this manner, the two parameters T c and λ are sufficient to determine the free energy due to the spins. Figure 1  to the G-L Hamiltonian describes data at temperatures ranging from 10 to 40 K quite well. However, in the critical region, discrepancies are substantial and significantly larger for the sample with higher donor concentration. The deficiency of the Landau approximation is its insensitivity to the short-range ordering, which smears out the phase transition. To estimate the amplitude of this effect, we treated the external magnetic field as an adjustable parameter ′ B and verified that its value of 3 T can significantly improve the description of the data for the sample with lower donor concentration (dashed lines in figure 1). For the sample with the higher donor concentration, the same procedure fails, even with a higher value of the correction field ′ = B 4 T. Assuming now, that we can account for the contribution due to the short-range ordering by introducing a correction field ′ B , which simply adds to the external magnetic field, we can account for the remaining contribution to the magnetization of the sample M as caused by the presence of BMPs. Thus, the presence of an extra field ′ B mimics the effect of a short-range order or other higher-order effects. It is rewarding that such a simple extra factor is sufficient to rationalize the data on a semi-quantitative level.
Having determined the values of the spin-functional parameters for the host magnetic semiconductor, we need to establish the values of parameters related to the impurity state. Most interesting for us is the neutral donor concentration   solid line) taken from [26]. The calculation of − n has been carried out using (B.6) with two approaches describing M 0 namely, either minimizing the system free energy (open circles) or using the Brillouin function for ferromagnetic case (dotted line). It is apparent from figure 2 that the double-dome structure of − n T ( ) is caused by the exchange contribution to the donor energy, because for the situation with = J 0 c , we obtain a monotonic curve (dashed line). This composes the signature of the BMP in ferromagnetic semiconductors. Our present approach underestimates the contribution due to the s-f exchange.
Following [26], we assume that the effective mass = m m * 0.15 0 and the static dielectric constant is ε = 4, which leads to the effective donor Bohr radius ≈ a 1.4 B nm. Thus, there are 400 Gd 3+ cations within the BMP radius in GdN. We can now calculate the system free energy originating from the presence of BMPs. Explicitly, it has the form   figure 3 with the experimental data for EuO of [25] (see figure 2 in [25]). The polaronic contribution above T c is clearly visible for > J 0 c in both figures and thus, our model also provides the double-dome feature for the free energy curve caused by the formation of BMPs. In figure 5, we plotted the calculated dependence F on M 0 for various temperatures. The Mexican hat shape signaling the onset of the magnetic order in the BMP starts developing below = T 135 K. However, the magnitude of this effect is low for temperatures in that regime. Nevertheless, within the assumed value of the model parameters, the equilibrium magnetization density is small but clearly non-zero, even at 130 K. The evolution of the Mexican hat feature is in the last situation gradual and effectively, the polaron  In this work, we have essentially extended the theory [11] valid for the single BMP in a paramagnetic DMS to the situation of a ferromagnetic semiconductor. This has been completed by including in the system effective Landau spin Hamiltonian the quartic term. A non-zero magnetization within the BMP cloud persists to temperatures well above T c and is in agreement with the recent experimental data reported by Natali et al [26] for GdN, as well as with the previous theoretical investigations on other ferromagnetic semiconductors [10,39]. The present continuous-medium approximation is effective because of two factors. First, the size of the BMP is relatively large (≈1.5 nm) and encompasses ≈ N 400 + Gd 3 spins of magnitude = S 7 2. Previously [15], it has been shown for the case with Gaussian fluctuations, that the spin cloud composed of > N 4 spins of magnitude = S 5 2 is represented already rather well by continuous distribution of exchange field acting on the donor electron. Second, the spin magnitude = S 7 2 is the largest possible and makes the quantum aspects of fluctuations negligible. In general, the thermal behavior of the spin splitting Δ T ( ), demonstrates once more [42] the influence of the thermodynamic fluctuation on a quantum (donor in the present case) state of individual electron. However, inclusion of a number of polarons − − n n d must be assumed, as well as an extra ′ B field, which is required to account quantitatively the data for small n d .
One should note that, in this formulation, we have not tackled explicitly the most general case of non-Gaussian fluctuations starting from the effective Hamiltonian   + e S , which is given by equation (5) and  S by equation (8). The splitting (6) plays the role of the inhomogeneous field acting on the spins. To tackle such a problem, a generalization of the standard renormalization group approach [40] would be required. Only then one can seriously analyze the strongly polarized BMP states for ≪ T T c such as those that appear in EuO and related magnetic semiconductors [2][3][4], where the magnetic contribution to the BMP binding energy becomes predominant.
Therefore, the Heisenberg spin-spin interaction takes the form: if we take only the interactions between z nearest neighbors, and a 0 is the lattice parameter. We assume additionally that, as we approach the phase-transition point,  Strictly speaking, in the continuous-medium approximation, F N represents a free energy density. Therefore, the total free energy functional in a space of d dimensions is where v 0 is the volume per spin (in the simple cubic case = v a 0 0 3 ). Integrating the third term by parts and neglecting the surface term (not always feasible), we obtain must now be added to equation (A.6). Additionally, we consider the fourthorder contribution in β = x J S z 0 . We then follow all of the steps that lead from equation (A.6) to equation (A.9) to also obtain the quartic term in a explicit form: