Quantitative completion and experimental test of Abrikosov's spin-glass theory for semiconductor hosts: the $y\sim1$ regime, realistic band structure, commensuration and dipolar crossovers, and dilute Eu$_x$Sr$_{1-x}$S data

A. A. Abrikosov (Journal of Low Temperature Physics 39, 217 (1980)) developed the theory of a spin glass whose localized impurity spins interact through the Bloembergen-Rowland mechanism - indirect exchange mediated by virtual electron-hole excitations across the band gap of a semiconductor host - giving an exponentially screened, oscillatory interaction $e^{-\alpha r}r^{-5/2}\sum_i\cos(\mathbf{K}_i\cdot\mathbf{r})$ in place of the metallic RKKY form. The theory derives the low-temperature thermodynamics from the distribution $\rho(q)$ of the oscillatory factor (logarithmic or power-logarithmic laws for the susceptibility $\chi$ and specific heat $C$ selected by the geometry of the band extrema), predicts a spin-glass scale $T_0=V_0S\alpha^{5/2}y^{3/2}e^{-y^{-1/2}}$ with $y=4\pi n_m/\alpha^3$, and fits three dilute Eu$_x$Sr$_{1-x}$S data points with $\alpha\sim10^7$ cm$^{-1}$. Its derivation, however, requires $y\ll1$ while the fitted data have $y=0.85$-$3$; it assumes isotropic parabolic bands; its commensurate-case and dipolar-crossover statements are qualitative; and its experimental test rests on a three-parameter fit to three points. The subsequent record leaves the following open: the direct follow-up line (Liu, Lee, Yu and co-workers, 1980-1992) recomputed the indirect exchange in semiconductors and diluted magnetic semiconductors with more realistic bands but did not redo the spin-glass statistics, the $y\sim1$ regime, or the Eu$_x$Sr$_{1-x}$S confrontation; the dilute-Eu$_x$Sr$_{1-x}$S experimental literature converged on a different microscopic picture (short-range ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor exchange, superparamagnetic pair blocking, dipolar anisotropy) without refuting or completing Abrikosov's program; and the modern Bloembergen-Rowland revival (diluted III-V semiconductors, GaN:Gd, gapped graphene, topological insulators) treats different systems and ordered or pair physics. A precise open core survives: the $y\sim1$ thermodynamics of the Bloembergen-Rowland exchange, the realistic-band form of $\rho(q)$ and of the preexponential factors, the quantitative commensuration and exchange-to-dipolar crossovers, and the still-unperformed quantitative test against low-concentration Eu$_x$Sr$_{1-x}$S data.

Background

A spin glass is a disordered magnet in which randomly positioned magnetic moments interact through couplings of competing sign, so that no periodic arrangement can satisfy them all (frustration); on cooling, the moments freeze into a random, history-dependent configuration below a characteristic temperature $T_0$ without developing long-range order. In the canonical metallic spin glasses, such as dilute Cu-Mn alloys, the moments are coupled by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction - an indirect exchange mediated by the conduction electrons - which oscillates in sign with separation roughly as $\cos(2k_Fr)/r^3$, with $k_F$ the Fermi wave vector, because only electrons at the Fermi surface contribute. Abrikosov (1980) asked what replaces this mechanism when the host is a semiconductor: a crystal with no free carriers at low temperature but with an energy gap $\Delta$ between the filled valence band and the empty conduction band.

The answer is the Bloembergen-Rowland mechanism, also called interband polarization or virtual electron-hole exchange: each localized spin virtually excites an electron across the gap, and the polarized valence and conduction bands mediate a spin-spin coupling (Bloembergen and Rowland, 1955). In a metal this contribution is dominated by the RKKY term; in a gapped host it is the leading long-range coupling. For a model host with a parabolic valence band of isotropic effective mass $m_h$ whose maximum is at crystal momentum $\mathbf{K}=0$, and equivalent parabolic conduction-band minima of mass $m_e$ at momenta $\mathbf{K}_i$, Abrikosov derived the long-distance asymptotic interaction between impurity spins $\mathbf{S}_1$ and $\mathbf{S}_2$ separated by $r$:

$$\mathscr{H}_{12}=-\frac{\pi^{-5/2}\left(\alpha m^{*}\right)^{3/2}}{\left[2\left(m_e+m_h\right)\right]^{1/2}}\left(\frac{J}{n}\right)^{2}\,\mathbf{S}_1\cdot\mathbf{S}_2\;e^{-\alpha r}\,r^{-5/2}\sum_i\cos\left(\mathbf{K}_i\cdot\mathbf{r}\right),$$

where $J$ is the local exchange constant coupling a spin to the band electrons, $n$ is the host-atom density, $m^{*}=(m_e^{-1}+m_h^{-1})^{-1}$ is the reduced effective mass, and $\alpha=\sqrt{2\Delta(m_e+m_h)}$ (in units with $\hbar=1$) is the inverse interaction length set by the gap and the masses. The decisive difference from RKKY is the exponential screening $e^{-\alpha r}$: pairs with very weak coupling arise not from a power-law tail but from accidental small values of the oscillatory sum $q(\mathbf{r})=\sum_i\cos(\mathbf{K}_i\cdot\mathbf{r})$ at typical impurity separations.

The low-temperature thermodynamics of such a dilute magnet is controlled by the statistics of these weak pairs. Abrikosov built on the pair-statistics theory he had developed with Moukhin for spin glasses with nonmagnetic defects and exponentially decaying interactions (Abrikosov and Moukhin, 1978), in which the distribution $\rho(q)$ of the oscillatory factor over random impurity separations enters the specific heat $C$ and the susceptibility $\chi$ directly. For conduction-band minima on fourth-order (C4, Si-like) symmetry axes of the cubic host he obtained a finite value $\rho(0)=6/(\eta\pi^3)$, where $\eta$ is a geometric factor equal to 1 or 2, which yields the logarithmic law $\chi(T)\propto\ln(T_0/T)$; for minima on third-order (C3, Ge-like) axes the small-$q$ behavior diverges as $\rho(|q|\ll1)=\frac{1}{2\pi^3\eta}\left(\ln^2\frac{32\eta}{|q|}-\frac{\pi^2}{4}\right)$, which modifies the scaling to $C(T)\propto T\ln^2(T_0/T)$ and $\chi(T)\propto\ln^3(T_0/T)$. For a simplified model interaction $V_0\,\mathbf{S}_1\cdot\mathbf{S}_2\cos(Kr)\,e^{-\alpha r}\,r^{-5/2}$, a steepest-descent (saddle-point) evaluation of the pair statistics gives the spin-glass scale

$$T_0=V_0\,S\,\alpha^{5/2}\,y^{3/2}\,e^{-y^{-1/2}},\qquad y=\frac{4\pi n_m}{\alpha^3},$$

where $V_0$ collects the exchange amplitude, $S$ the impurity spin magnitude, and $n_m$ the volumetric density of magnetic impurities; the parameter $y$ counts impurities per interaction volume $\alpha^{-3}$, and the whole derivation assumes $y\ll1$ so that the saddle point is well separated from the integration boundaries. When the impurities occupy definite crystallographic sublattices of an fcc-based (face-centered-cubic) lattice and the minima $\mathbf{K}_i$ are commensurate with the reciprocal lattice, the sum $q(\mathbf{r})$ takes only discrete values, $\rho(q)$ collapses to a sum of delta functions (for example $\frac{1}{4}\delta(q-3)+\frac{3}{4}\delta(q+1)$), the glass splits into quasi-independent interpenetrating subsystems of alternating coupling sign that are weakly coupled by magnetic dipole-dipole interactions (which fall off as $r^{-3}$ and depend on the orientation of the moments), and the susceptibility is predicted to show two maxima.

The material system is Eu$_x$Sr$_{1-x}$S. EuS is a ferromagnetic semiconductor with the rocksalt structure, with Eu$^{2+}$ ions (spin $S=7/2$) on the fcc cation sublattice; substituting nonmagnetic Sr$^{2+}$ dilutes the magnet, which becomes a canonical insulating spin glass over $0.13\lesssim x\lesssim0.65$ (Maletta and Felsch, 1979; Maletta, 1980). Abrikosov fitted the measured concentration dependence of $T_0$ at $x=0.0085$, $0.017$, and $0.03$ with the three-parameter form $\ln T_0=a+b\ln y-y^{-1/2}$ with $y=dx$, obtaining $d=100$; identifying $y=4\pi n_m/\alpha^3$ with $n_m=xn$ and $n\sim10^{22}$ cm$^{-3}$ gives $\alpha\sim10^7$ cm$^{-1}$, consistent with $\Delta\sim1$ eV and $m_e,m_h\sim0.1m_0$ (with $m_0$ the free-electron mass). But the fitted range corresponds to $y=0.85$-$3$, far outside the $y\ll1$ regime of the derivation - a limitation the paper itself acknowledges. The standard microscopic picture of the europium chalcogenides offers a competing mechanism at these concentrations: short-range exchange between nearest-neighbor (nn) Eu$^{2+}$ that is ferromagnetic ($J_1$, with $J_1/k\approx0.1$ K measured by electron-spin resonance, the microwave absorption of the impurity spins) and between next-nearest-neighbor (nnn) Eu$^{2+}$ that is antiferromagnetic ($J_2$) (Deville et al., 1981; Maletta, 1980).

Problem Statement

The problem, drawn from the open ends of the program of Abrikosov (1980), is to make the theory of a spin glass whose impurity spins interact through semiconductor-mediated (Bloembergen-Rowland) exchange quantitative at the experimentally relevant concentrations and to confront it with data. The open program comprises four directions, each with its own quantitative resolution:

(1) Regime of the dilution parameter. The thermodynamic formulas, including $T_0=V_0S\alpha^{5/2}y^{3/2}e^{-y^{-1/2}}$ and the $C(T)$ and $\chi(T)$ laws, were derived by steepest descent under the condition $y=4\pi n_m/\alpha^3\ll1$, while the Eu$_x$Sr$_{1-x}$S data used sit at $y\sim1$. Establish the spin-glass thermodynamics of the same interaction at $y\sim1$: what the corrected concentration and temperature dependences of $T_0$, $C$, and $\chi$ are, including the validity of the exponential factor $e^{-y^{-1/2}}$ outside the asymptotic regime.

(2) Realistic band structure. The exchange was derived assuming isotropic, parabolic band extrema and neglecting full Brillouin-zone structure (the complete band geometry of the crystal's reciprocal-space cell). Determine how nonparabolic dispersion, anisotropic effective masses, multiple inequivalent or spin-orbit-split extrema, and higher-energy bands each modify the interaction amplitude and inverse length $\alpha$ - hence the preexponential factors of $T_0$, $C$, and $\chi$ - and the small-$q$ behavior of the distribution $\rho(q)$ - hence the boundary between the logarithmic and power-logarithmic thermodynamic laws.

(3) Commensuration and dipolar crossovers. The commensurate case (delta-function $\rho(q)$, quasi-independent sublattice subsystems) and the role of longer-range dipolar interactions at weak-exchange concentrations and separations were discussed only qualitatively. Compute the quantitative temperature and concentration windows over which (a) commensuration produces the predicted two-maximum susceptibility before incommensurate statistics take over, and (b) dipole-dipole interactions compete with or dominate the exponentially screened exchange, including the predicted saturation of $\chi$ from the effective fields of distant neighbors.

(4) Experimental confrontation. Test the theoretical forms - especially the preexponential factors, not only the leading exponential - against the available low-concentration Eu$_x$Sr$_{1-x}$S data and the best independently available band-structure input, and refine the extracted $\alpha$; the original fit used three data points and three free parameters and so constrains nothing by itself.

References

1. M. A. Ruderman and C. Kittel, Indirect exchange coupling of nuclear magnetic moments by conduction electrons, Physical Review 96, 99-102 (1954), DOI 10.1103/PhysRev.96.99, https://doi.org/10.1103/PhysRev.96.99
2. Tadao Kasuya, A theory of metallic ferro- and antiferromagnetism on Zener's model, Progress of Theoretical Physics 16, 45-57 (1956), DOI 10.1143/PTP.16.45, https://doi.org/10.1143/PTP.16.45
3. Kei Yosida, Magnetic properties of Cu-Mn alloys, Physical Review 106, 893-898 (1957), DOI 10.1103/PhysRev.106.893, https://doi.org/10.1103/PhysRev.106.893
4. N. Bloembergen and T. J. Rowland, Nuclear spin exchange in solids: Tl$^{203}$ and Tl$^{205}$ magnetic resonance in thallium and thallic oxide, Physical Review 97, 1679-1698 (1955), DOI 10.1103/PhysRev.97.1679, https://doi.org/10.1103/PhysRev.97.1679
5. A. A. Abrikosov and S. I. Moukhin, Spin-glass with nonmagnetic defects, Journal of Low Temperature Physics 33, 207-229 (1978), DOI 10.1007/BF00114995, https://doi.org/10.1007/BF00114995
6. A. A. Abrikosov, Spin-glass with short-range interaction in a magnetic field, Journal of Low Temperature Physics 33, 505-520 (1978), DOI 10.1007/BF00115572, https://doi.org/10.1007/BF00115572
7. A. A. Abrikosov, One-dimensional spin-glass, Journal of Low Temperature Physics 37, 111-122 (1979), DOI 10.1007/BF00114060, https://doi.org/10.1007/BF00114060
8. A. A. Abrikosov, Spin-glass with a semiconductor as host, Journal of Low Temperature Physics 39, 217-229 (1980), DOI 10.1007/BF00118075, https://doi.org/10.1007/BF00118075
9. H. Maletta and W. Felsch, Insulating spin-glass system Eu$_x$Sr$_{1-x}$S, Physical Review B 20, 1245-1260 (1979), DOI 10.1103/PhysRevB.20.1245, https://doi.org/10.1103/PhysRevB.20.1245
10. J. L. Tholence, F. Holtzberg, H. Godfrin, H. V. Löhneysen and R. Tournier, Blocking of pairs and 'spin glass freezing' in Eu$_x$Sr$_{1-x}$S, Journal de Physique Colloques 39, C6-928-C6-930 (1978), DOI 10.1051/jphyscol:19786412, https://doi.org/10.1051/jphyscol:19786412
11. J. L. Tholence, Spin-glass versus 'blocking' in dilute Eu$_x$Sr$_{1-x}$S, Journal of Applied Physics 50, 7369-7371 (1979), DOI 10.1063/1.326945, https://doi.org/10.1063/1.326945
12. G. Eiselt, J. Kötzler, H. Maletta, D. Stauffer and K. Binder, Magnetic 'blocking' in very dilute Eu$_x$Sr$_{1-x}$S: Experiment versus theory, Physical Review B 19, 2664-2676 (1979), DOI 10.1103/PhysRevB.19.2664, https://doi.org/10.1103/PhysRevB.19.2664
13. J. Kötzler and G. Eiselt, Observation of spin-cluster freezing in dilute (Eu$_x$Sr$_{1-x}$)S by low-frequency magnetic absorption, Journal of Physics C: Solid State Physics 12, L469-L474 (1979), DOI 10.1088/0022-3719/12/12/004, https://doi.org/10.1088/0022-3719/12/12/004
14. G. Eiselt and J. Kötzler, Magnetization dynamics in very dilute (Eu$_x$Sr$_{1-x}$)S, Journal of Magnetism and Magnetic Materials 15-18, 191-192 (1980), DOI 10.1016/0304-8853(80)91011-2, https://doi.org/10.1016/0304-8853(80)91011-2
15. D. Meschede, F. Steglich, W. Felsch, H. Maletta and W. Zinn, Specific heat of insulating spin-glasses (Eu,Sr)S near the onset of ferromagnetism, Physical Review Letters 44, 102-105 (1980), DOI 10.1103/PhysRevLett.44.102, https://doi.org/10.1103/PhysRevLett.44.102
16. H. Maletta, Spin-glass properties of (Eu, Sr)S, Journal de Physique Colloques 41, C5-115-C5-125 (1980), DOI 10.1051/jphyscol:1980521, https://doi.org/10.1051/jphyscol:1980521
17. A. Deville, C. Arzoumanian, B. Gaillard, C. Blanchard, J. P. Jamet and H. Maletta, Electron resonance in the insulating spin glass Eu$_{0.4}$Sr$_{0.6}$S, Journal de Physique 42, 1641-1646 (1981), DOI 10.1051/jphys:0198100420120164100, https://doi.org/10.1051/jphys:0198100420120164100
18. J. Kötzler and G. Eiselt, Possibility of dipolar spin-glass in very dilute (Eu$_x$Sr$_{1-x}$)S, Physical Review B 25, 3207-3214 (1982), DOI 10.1103/PhysRevB.25.3207, https://doi.org/10.1103/PhysRevB.25.3207
19. L. Liu, Indirect exchange interaction in semiconductors, Solid State Communications 35, 187-188 (1980), DOI 10.1016/0038-1098(80)90241-0, https://doi.org/10.1016/0038-1098(80)90241-0
20. Ven-Chung Lee and L. Liu, Indirect exchange interaction in semiconductors, Journal of Physics C: Solid State Physics 16, L1087-L1090 (1983), DOI 10.1088/0022-3719/16/30/003, https://doi.org/10.1088/0022-3719/16/30/003
21. Ven-Chung Lee and L. Liu, Indirect interaction in semiconductors, Solid State Communications 48, 341-344 (1983), DOI 10.1016/0038-1098(83)90736-6, https://doi.org/10.1016/0038-1098(83)90736-6
22. L. Liu, Magnetic interactions in europium compounds, Solid State Communications 46, 83-85 (1983), DOI 10.1016/0038-1098(83)90037-6, https://doi.org/10.1016/0038-1098(83)90037-6
23. Ven-Chung Lee and L. Liu, Magnetic interaction in europium chalcogenides, Solid State Communications 48, 795-797 (1983), DOI 10.1016/0038-1098(83)91020-7, https://doi.org/10.1016/0038-1098(83)91020-7
24. Ven-Chung Lee and L. Liu, Indirect exchange interaction in dilute magnetic semiconductors, Physical Review B 29, 2125-2130 (1984), DOI 10.1103/PhysRevB.29.2125, https://doi.org/10.1103/PhysRevB.29.2125
25. Steven J. Frisken and D. J. Miller, Indirect-exchange interaction between localized moments in semiconductors, Physical Review B 33, 7134-7138 (1986), DOI 10.1103/PhysRevB.33.7134, https://doi.org/10.1103/PhysRevB.33.7134
26. Shinn-Sheng Yu and Ven-Chung Lee, Indirect exchange interaction in semimagnetic semiconductors, Physical Review B 40, 10621-10624 (1989), DOI 10.1103/PhysRevB.40.10621, https://doi.org/10.1103/PhysRevB.40.10621
27. Shinn-Sheng Yu and Ven-Chung Lee, Indirect exchange interaction in diluted magnetic semiconductors, Journal of Physics: Condensed Matter 4, 2961-2975 (1992), DOI 10.1088/0953-8984/4/11/021, https://doi.org/10.1088/0953-8984/4/11/021
28. G. Bastard and C. Lewiner, Indirect exchange interaction in parabolic symmetry-induced zero gap semiconductors, Journal de Physique Lettres 40, L189-L192 (1979), DOI 10.1051/jphyslet:01979004008018900, https://doi.org/10.1051/jphyslet:01979004008018900
29. L. Liu and G. Bastard, Indirect exchange interaction in lead salts, Physical Review B 25, 487-489 (1982), DOI 10.1103/PhysRevB.25.487, https://doi.org/10.1103/PhysRevB.25.487
30. S. V. Iordanskii, F. G. Pikus and G. E. Pikus, Indirect exchange interaction for zero-gap semiconductors in the Kane model, Journal of Applied Physics 76, 424-431 (1994), DOI 10.1063/1.357092, https://doi.org/10.1063/1.357092
31. K. C. Hass, B. E. Larson, H. Ehrenreich and A. E. Carlsson, Magnetic interactions in diluted magnetic semiconductors, Journal of Magnetism and Magnetic Materials 54-57, 1283-1284 (1986), DOI 10.1016/0304-8853(86)90819-X, https://doi.org/10.1016/0304-8853(86)90819-X
32. M. Crişan, Zs. Gulácsi and M. Gulácsi, Spin glass with short-range interaction induced by an electromagnetic field in semiconductors, Journal of Magnetism and Magnetic Materials 38, 67-77 (1983), DOI 10.1016/0304-8853(83)90104-X, https://doi.org/10.1016/0304-8853(83)90104-X
33. Zs. Gulácsi, M. Gulácsi and M. Crisan, A possible mechanism for the short-range spin-glass state in insulators, Journal of Magnetism and Magnetic Materials 40, 247-258 (1984), DOI 10.1016/0304-8853(84)90315-9, https://doi.org/10.1016/0304-8853(84)90315-9
34. R. Skomski, J. Zhou, J. Zhang and D. J. Sellmyer, Indirect exchange in dilute magnetic semiconductors, Journal of Applied Physics 99, 08D504 (2006), DOI 10.1063/1.2159394, https://doi.org/10.1063/1.2159394
35. V. I. Litvinov and V. K. Dugaev, Ferromagnetism in magnetically doped III-V semiconductors, Physical Review Letters 86, 5593-5596 (2001), DOI 10.1103/PhysRevLett.86.5593, https://doi.org/10.1103/PhysRevLett.86.5593
36. V. K. Dugaev, V. I. Litvinov, J. Barnas and M. Vieira, A new mechanism of exchange interaction in ferromagnetic semiconductors, arXiv:cond-mat/0203119, https://arxiv.org/abs/cond-mat/0203119
37. V. K. Dugaev, V. I. Litvinov and J. Barnas, Exchange interaction of magnetic impurities in graphene, arXiv:cond-mat/0610658, https://arxiv.org/abs/cond-mat/0610658 (published version: Physical Review B 74, 224438 (2006), DOI 10.1103/PhysRevB.74.224438, https://doi.org/10.1103/PhysRevB.74.224438)
38. V. I. Litvinov and V. K. Dugaev, Room-temperature ferromagnetism in dielectric GaN(Gd), arXiv:0905.0500, https://arxiv.org/abs/0905.0500 (published version: Applied Physics Letters 94, 212506 (2009), DOI 10.1063/1.3143670, https://doi.org/10.1063/1.3143670)
39. V. I. Litvinov, Magnetic exchange interaction in topological insulators, Physical Review B 89, 235316 (2014), DOI 10.1103/PhysRevB.89.235316, https://doi.org/10.1103/PhysRevB.89.235316
40. V. I. Litvinov, Oscillating Bloembergen-Rowland interaction in three-dimensional topological insulators, Physical Review B 94, 035138 (2016), DOI 10.1103/PhysRevB.94.035138, https://doi.org/10.1103/PhysRevB.94.035138
41. Vladimir Litvinov, Exchange interaction in semiconductors and metals, chapter in Wide Bandgap Semiconductor Spintronics (Jenny Stanford Publishing, 2016), pp. 91-113, DOI 10.1201/b20038-8, https://doi.org/10.1201/b20038-8
42. Vladimir Litvinov, Indirect exchange interaction mediated by Dirac fermions, chapter in Magnetism in Topological Insulators (Springer, Cham, 2019), DOI 10.1007/978-3-030-12053-5_7, https://doi.org/10.1007/978-3-030-12053-5_7
43. Xiaofei Fan, Guodong Wei, Xiaoyang Lin, Xinhe Wang, Zhizhong Si, Xueying Zhang, Qiming Shao, Stéphane Mangin, Eric Fullerton, Lei Jiang and Weisheng Zhao, Phase-change control of interlayer exchange coupling, arXiv:1907.10784, https://arxiv.org/abs/1907.10784 (published version: Reversible switching of interlayer exchange coupling through atomically thin VO$_2$ via electronic state modulation, Matter 2, 1582-1593 (2020), DOI 10.1016/j.matt.2020.04.001, https://doi.org/10.1016/j.matt.2020.04.001)
44. K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Reviews of Modern Physics 58, 801-976 (1986), DOI 10.1103/RevModPhys.58.801, https://doi.org/10.1103/RevModPhys.58.801