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Disorder and frustration in heavy-fermion compounds
Physica B 259—261 (1999) 882—886
Disorder and frustration in heavy-fermion compounds J.A. Mydosh* Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, Netherlands
Abstract Quantum phase transitions represent the central theme of current research in strongly interacting condensed matter physics. When disorder and frustration are incorporated into the problem, new features are predicted by theory that have not been fully examined experimentally. This brief review will consider the effects of disorder and frustration on the magnetic properties of two heavy-fermion materials, UNi B and URh Ge . The first, possessing only frustration in its hexagonal lattice, exhibits a unique ground state in which 2/3 of the U-moments form two different (vortex and pairs) X—½ structures while the remaining 1/3 appear frustrated, i.e., do not order down to 0.5 K. URh Ge possesses both disorder and frustration and shows a spin-glass freezing at 9 K with the U-moments randomly aligned up/down along the tetragonal c-axis, i.e., Ising-like. We shall illustrate the possibilities that these systems are close to a ¹"0 quantum critical point which governs their unusual finite temperature behaviors. 1999 Elsevier Science B.V. All rights reserved. Keywords: Heavy-fermion compounds; UNi B; URh Ge ; Disorder and frustration
1. Introduction Magnetism in strongly correlated electron systems remains an intriguing and often surprising research area (See for example the collection of papers in Ref. [1]). This is especially true of the heavy-fermion materials usually based on Ce or U intermetallic compounds. What is particularly interesting here are not only the unusual forms of quantum magnetism which often appear but also their coexistence with superconductivity and structural transformations. By quantum magnetism we mean the realm where intrinsic quantum effects play a unique and dominant role. For example, a quantum spin directly governs the magnetic properties, or at low-temperatures zeropoint fluctuations drive a new and completely different critical phenomenon without thermal fluctuations. In addition, tunneling or coherence effects can strongly couple the systems and give rise to macroscopic quantum
* Fax: 071-275404; e-mail: [email protected].
behaviors. New terms and their associated concepts are constantly appearing in the literature as for instance: quantum spins, charge and spin gaps, quantum fluctuations and frustration, ¹"0 phase transitions or quantum critical points, quantum disorder and spin glasses, spin liquid, resonanting valency bonds (RVB), spincharge separation, stripes, non-Fermi liquids, Kondo insulators, etc. Due to the difficulties of obtaining wellcharacterized samples and performing measurements at very low temperatures, experiment is sadly lacking and has fallen far behind the theoretical endeavors. In this paper we focus upon two strongly correlated uranium intermetallic compounds recently fabricated in single crystal form, UNi B and URh Ge , which ex hibit unusual magnetic properties related to the presence of disorder and frustration. Their special behavior places them in the above realm of quantum magnetism and they may serve as “test-beds” for the various theoretical predictions invoking quantum disorder and frustration. Before reviewing the experimental properties of these compounds and comparing them with recent theories, let
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 9 1 5 - 6
J.A. Mydosh / Physica B 259—261 (1999) 882—886
883
Fig. 1. Possible magnetic ground states obtainable from the perspective of site disorder and frustration.
us first try to gain some perspective regarding heavyfermion magnetism with respect to site disorder and magnetic frustration. Consider the 2;2 matrix of possible magnetic ground states shown in Fig. 1 and usually applied to insulating compounds. If we now select four representative heavy-fermion systems, we can nicely fill in the various squares with different combinations (or degree) of disorder and/or frustration — all leading to unconventional magnetic behavior of the given material. URu Si — possessing no disorder or frustration, yet an unusual “small-moment” k +0.03 l magnet ism appears via a sharp phase transition below 17 K. Superconductivity then coexists with this state at about 1 K. In spite of almost fifteen years of investigation, even now there is no agreement or “smoking gun” regarding the 17 K transition. According to the bulk properties and their field dependences, the “ordering” is related to a CDW or an antiferromagnetic array of quadrupoles [2]. Yet the definitive localprobe experiment (high-resolution X-ray diffraction or NMR) is still lacking. (ii) CePd Al — possessing disorder where single crys tals exhibit a strange multi-site randomness on the Al sublattice. Magnetic frustration is not present, however, the disorder prevents the expected longrange antiferromagnetic state. In polycrystalline samples the randomness is surprisingly less and the antiferromagnetic state does indeed form [3,4].
Fig. 2. (a) Crystallographic CeCo B-type subcell of UNi B. (b) Magnetic structure of hexagonal UNi B projected onto the basal plane. The magnetic layers are stacked ferromagnetically along the c-direction. The thin solid line represents the magnetic unit-cell. Note the presence of the “frustrated” U-spins indicated by (1) and (2) with two inequivalent magnetic environments. The nn and nnn magnetic exchanges are indicated by J and J . From Mentink et al. [8].
(i)
Due to lack of space we shall not discuss these interesting systems any further (for additional discussion see Ref. [5]) and move onto the other two compounds in Fig. 1, UNi B and URh Ge , and their quantum-magnetism behaviors.
2. UNi4B This material possesses only geometric frustration because of its hexagonal lattice and in-plane antiferromag-
netic interactions. As far as our metallurgical, and X-ray and neutron-diffraction investigations [6,7] can tell, there is no site disorder in this single-crystal compound, probably because of large size differences in the three atomic constituents. Magnetically a sharp phase transition occurs at ¹ "20 K which represents a partial anti, ferromagnetic ordering of the U-moments. Fig. 2 depicts the results of magnetic neutron-scattering experiments [8]. The most probable spin structure is sketched for the ordered U-moments (k "1.2 l at ¹"0) which are confined to the hexagonal basal plane. Note that only 2/3 of the U-ions participate in the long-range order. The remaining 1/3 are free (paramagnetic) or frustrated (spin liquid) or compensated (Kondo) spins that dominate the bulk properties at low temperatures (¹;¹ ). The long, range ordered structure is most unusual for a 2D triangular lattice (all the spins are simply ferromagnetically coupled along the c-axis). Two types of alignments surround two different sites of frustrated spin, labeled (1) and (2) in Fig. 2. But how can this occur if all the U-sites are equivalent as drawn in the figure? Answer: they are not. There is a small superstructure distortion of the crystal lattice that has been observed via X-ray and neutron diffraction. These measurements show that the surrounding U (along with the Ni and B) atoms to (1) and (2) displace in such a manner so as to slightly modify their local symmetry leaving that at U-sites (1) and (2) unchanged. Thus, these sites are “marked” by the crystallographic superstructure and this designates them as the
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centers of the unique vortex-like arrangement at (1) and the encircling three pairs at (2). For a complete description of both the magnetic and lattice structures, see Ref. [6,7]. With respect to a quantum behavior or phase transitions, the question remains as to what happens to these free, frustrated or compensated spins when the UNi B is cooled to ¹P0. Fig. 3 exhibits the low-temperature specific heat (plotted as C/¹) and the resistivity (o and do/d¹) [9]. Clearly there is a subtle transformation taking place at 0.3 K. What is exceptional here is that the c-value (,C/¹) is much more sensitive to the transformation than o(¹) or do/d¹. The latter only changes by a few percent with a broad maximum in its derivative, see Fig. 3. However, the entropy under the C/¹ versus ¹ curve to 1 K is much smaller than that if the remaining 1/3 U moments ordered. Additional measurements (not shown) of the AC-susceptibility and lSR seem to indicate no alteration in the magnetic state, i.e., there is neither a sharp magnetic response nor a rapid growth of internal fields around 0.3 K [10]. At this time we can only speculate as to the cause of this low-temperature anomaly. The behavior seems to represent a crossover either to a reorientation of the ordered U-moments spin structure, or more likely, to a new frozen quantum state of the frustrated spins. An interpretation in terms of the Kondo effect appears improbable because c does not become constant in the low ¹ limit (¹;¹ ) but instead ) traverses a maximum and continues to decrease. Furthermore, the resistivity has a clear positive slope in this ¹-regime. Two recent theoretical models have been applied to UNi B. The first by Lacroix et al. [11] treats its magnetic properties in terms of first and second-nearest-neighbor exchanges, J and J , and a Kondo-effect parameter D. By forming an effective Hamiltonian using these three parameters and minimizing its energy, the magnetic phase diagram can be obtained in parameter space. For reasonable values of J , J and D, the magnetic structure, as shown from neutron diffraction in Fig. 2, appears in the calculation. The Kondo effect is reasoned to play a key role by selectively compensating the 1/3 U-magnetic moments at sites (1) and (2). Without spin these moments cannot enter into the ordered structure and remain independent of it. However, in the process of being compensated these U-spins should act as Kondo impurities exhibiting the behavior of such, e.g., a negative temperature coefficient of resistivity, an enhanced and constant c-value at low ¹, and a Curie—Weiss-like susceptibility which also becomes constant at low ¹. Preliminary measurements given in Fig. 3, and the susceptibility, which exhibits a small broad maximum around 0.3 K, are inconsistent with a simple Kondo compensation of the frustrated U-moments. The second theory by Tejima and Oguchi [12] uses a Heisenberg Hamiltonian with various pairs of nn and
Fig. 3. Low-temperature specific heat of UNi B plotted as C/¹ versus ¹. Inset: resistivity o and do/d¹ in the same ¹-region. From Movshovich et al. [9].
nnn interacting spins in the X-½ plane; also involved is an interhexagon exchange coupling. Minimizing the (classical) energy of the Hamiltonian determines the spin configuration of the classical ground state. Indeed for proper choice of exchange parameters a spin configuring agreeing with the experimental one in Fig. 2 can be obtained, if quantum fluctuations are included to unstabilize the standard 120° (3 sublattice) Ne´el order. These fluctuations, therefore, lead to 1/3 of the U-moments remaining paramagnetic. Tejima and Oguchi have also calculated the low ¹-dependence of the specific heat. Their results are that spin waves give the dominant contribution and that C/¹ reaches a broad maximum at +2 K and smoothly decreases to zero below. This behavior is not in agreement with experiment. And so both available theories do not seem to confront or describe the unusual low ¹ behavior of UNi B. 3. URh2Ge2 Here we consider a situation where both lattice disorder and magnetic frustration are present. But how can a perfectly stoichiometric, single crystal intermetallic compound be disordered? We have already mentioned the subtle multi-sites of the Al-atoms in CePd Al that destroy the long-range antiferromagnetic order. However, for URh Ge the randomness is more dramatic and because of the randomly oscillating exchange coupling (RKKY), magnetic frustration results and a clear spinglass transition occurs. Now what is the exact cause of the disorder? Based upon extensive metallurgical, and neutron and X-ray diffraction measurements [13] we conclude: (i) there is a mixture of the two possible bct-crystal structures, ThCr Si and CaBe Ge , (ii) a random exchange of Rh and Ge sites occurs, (iii) a distribution of the free positional z-parameter of Rh and Ge takes place, and
J.A. Mydosh / Physica B 259—261 (1999) 882—886
(iv) a small amount of vacancies exists on these ligand sites. Note the magnetic U-sites are perfectly ordered, only the local environments are affected by the Rh and Ge randomness. When the above is combined with predominantly ferromagnetic interactions in the basal plane and antiferromagnetic couplings along the c-axis, we have the competing exchanges and necessary frustration to create a 3D, Ising-like, random-bond, metallic, ‘‘heavy-fermion’’ spin glass — the first of its kind [14]. Fig. 4 shows the real and imaginary parts of the linear AC-susceptibility at different frequencies u. Note the sharp u-dependent cusp of s around 10 K that shifts to higher temperatures as the frequency is increased. Also remarkable is that s for B (the driving AC field) parallel to the c-axis is much larger than s for B#aˆ . This indicates an Ising character. s appears as a much smaller step whose point of inflection corresponds to the cusp in s , and accordingly, designates the spin-glass freezing temperature ¹ (u). The frequency shift of ¹ , *¹ /(¹ *log u) is 0.025, a value typical for metallic spin glass. The DC-susceptibility, s "M/H, (not shown) also exhibits "! the characteristic behavior of an archetypal spin glass, namely, after zero-field cooling, a cusp is formed in s at "! ¹ , while upon field cooling a plateau develops below ¹ . There are further the usual irreversibilities and logarithmic time dependences when the external field is modified at ¹(¹ . The magnetic specific heat C resembles that of a canonical spin glass. C is featureless below 10 K, and a broad maximum appears in C/¹ above. At low ¹, C varies between ¹ and ¹ with the linear ¹ coefficient, c, having a value of 130 mJ/mol K in the limit ¹P0. The above data, when taken collectively, demonstrate that URh Ge is a strongly correlated spin glass.
Fig. 4. Frequency dependence of the AC-susceptibility s and s for URh Ge with the driving field parallel to the a- and c-axis. The frequencies are 1.16, 11.6, 116 and 1160 Hz. From Su¨llow et al. [14].
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Only the electrical resistivity does not show the conventional properties of metallic spin glasses [15]. Fig. 5 plots the total resistivity o(¹) up to room temperature. Here there are (i) unusually large values of o exceeding the Ioffe—Regel criterion of +200 lX cm, (ii) a large and ¹-dependent anisotropy between the a- and c-axis, and (iii) negative temperature coefficients (do/d¹(0) for both crystallographic directions. The solid line in Fig. 5 represents a fit to (p #m(¹)\ where p O0 is the ¹"0 metallic conductivity and “m” a fit parameter. The (¹-temperature dependence, which has often been observed in metallic glasses, comes from “incipient localization” or corrections to the Boltzmann formula when the inelastic mean free path is much larger than the elastic mean free path for diffusive electronic transport in a disordered medium. Along the a-axis the above fit is quite good so there is little influence of spin or magnetic scattering. However, along the c-axis clearly an additional scattering contribution is found. If we separate the data into two components, viz., disorder and magnetic, and let the dashed line in Fig. 5 represent the former contribution, then a reasonable approximation for the magnetic part is gained by subtracting the dashed line from the experimental data. The difference is o (¹), #cˆ , which
exhibits a marked drop at about 10 K (+¹ ) and be comes constant (as it should) for ¹*60 K. Hence, we have developed a method to track ¹ via the resistivity along the Ising axis. What makes this system even more intriguing is its low freezing temperature and Ising character, and thereby, its proximity to a theoretically predicted quantum critical point. Recent theories (see for example Ref. [16]) have calculated a unique temperature versus quantum fluctuation strength, r, phase diagram for a spin glass — see Fig. 6. As ¹ is lowered by increasing r, a quantum critical regime is accessed whereby a non-Fermi liquid results (II) with unusual experimental behaviors of C/¹, s, o, 1/¹ , etc. The quantum critical point govern ing these finite ¹ properties exists at r"r and ¹ "0.
Fig. 5. Resistivity versus temperature for URh Ge along the a and c axes. See text for line fits. From Su¨llow et al. [15].
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type or in tiny amounts, can greatly modify the magnetic character especially if frustration also exists. At present it is extremely difficult to characterize the disorder using metallurgical technique and what little we know is usually based on indirect physical measurements. Finally, there remain vast open areas of U intermetallic compounds where new and unexpected phenomena can be found via various combinations of elements and structures. Fig. 6. Theoretical phase diagram for a metallic spin glass. For the case of URh Ge “r”, the tuning parameter, can be disorder, pressure or transverse (to Ising axis) magnetic field. See text for description of various regions. From Sachdev and Read [16].
Increasing r beyond r we reach a quantum-disordered region (I) with its own new and distinct characteristics different from the preceding regime (quantum critical). In the experimental situation for URh Ge the quantum fluctuation parameter r may be tuned by varying the disorder via the sample preparation, applying hydrostatic pressure or using an external transverse (to the Ising axis) magnetic field. Certainly one of these three should reduce ¹ P0 and make the region around the quantum critical point (see Fig. 6) available for measurement. This would also allow entrance to the proposed quantum Griffiths singularities [17,18] where strong divergences in the susceptibilities are expected. In addition, as we know prolong annealing the single-crystal samples removes the crystallographic disorder and a long-range ordered antiferromagnetic state appears at ¹ "15 K. Thus, we have , an extremely favorable temperature region (9 to 15 K) with which to study the classical Griffiths phase by its relaxation behavior of the magnetization. URh Ge is a very promising material for future experimentation and a number of the above phenomena are presently being explored. In conclusion, we have shown that disorder and frustration play a major role in determining the behavior of strong correlated material and lead to novel effects of quantum magnetism. Disorder, even of an unintentional
Acknowledgements I wish to acknowledge my long-standing collaborations with A.A. Menovsky and G.J. Nieuwenhuys and my former Ph.D. students S.A.M. Mentink and S. Su¨llow, who performed the major work on UNi B and URh Ge . References [1] J. Phys.: Condens. Matter 8 (1996) 9675—10148, Physica B 230—232 (1997) 1—1082. [2] P. Santini, Phys. Rev. B 57 (1998) 5191. [3] S.A.M. Mentink et al., Phys. Rev. B 49 (1994) 15759. [4] H. Tou et al., J. Phys. Soc. Japan 63 (1994) 4176. [5] J.A. Mydosh, Z. Phys. B 103 (1997) 251. [6] S.A.M. Mentink, Ph.D. Thesis, Leiden University, 1994. [7] A. Drost, Ph.D. Thesis, Leiden University, 1995. [8] S.A.M. Mentink et al., Phys. Rev. Lett. 73 (1994) 1031. [9] R. Movshovich et al., to be published. [10] M.W. Meisel, G.J. Nieuwenhuys, private communications. [11] C. Lacroix et al., Phys. Rev. Lett. 77 (1996) 526. Physica B 230—232 (1997) 529. [12] S. Tejima, A. Oguchi, J. Phys. Soc. Japan 66 (1997) 3611. [13] S. Su¨llow, Ph.D. Thesis, Leiden University, 1996. [14] S. Su¨llow et al., Phys. Rev. Lett. 78 (1997) 384. [15] S. Su¨llow et al., to be published. [16] S. Sachdev, N. Read, J. Phys.: Condens. Matter 8 (1996) 9723. [17] H. Rieger, A.P. Young, Phys. Rev. B 54 (1996) 3328. [18] M. Guo et al., Phys. Rev. 54 (1996) 3336.
Disorder and frustration in heavy-fermion compounds J.A. Mydosh* Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, Netherlands
Abstract Quantum phase transitions represent the central theme of current research in strongly interacting condensed matter physics. When disorder and frustration are incorporated into the problem, new features are predicted by theory that have not been fully examined experimentally. This brief review will consider the effects of disorder and frustration on the magnetic properties of two heavy-fermion materials, UNi B and URh Ge . The first, possessing only frustration in its hexagonal lattice, exhibits a unique ground state in which 2/3 of the U-moments form two different (vortex and pairs) X—½ structures while the remaining 1/3 appear frustrated, i.e., do not order down to 0.5 K. URh Ge possesses both disorder and frustration and shows a spin-glass freezing at 9 K with the U-moments randomly aligned up/down along the tetragonal c-axis, i.e., Ising-like. We shall illustrate the possibilities that these systems are close to a ¹"0 quantum critical point which governs their unusual finite temperature behaviors. 1999 Elsevier Science B.V. All rights reserved. Keywords: Heavy-fermion compounds; UNi B; URh Ge ; Disorder and frustration
1. Introduction Magnetism in strongly correlated electron systems remains an intriguing and often surprising research area (See for example the collection of papers in Ref. [1]). This is especially true of the heavy-fermion materials usually based on Ce or U intermetallic compounds. What is particularly interesting here are not only the unusual forms of quantum magnetism which often appear but also their coexistence with superconductivity and structural transformations. By quantum magnetism we mean the realm where intrinsic quantum effects play a unique and dominant role. For example, a quantum spin directly governs the magnetic properties, or at low-temperatures zeropoint fluctuations drive a new and completely different critical phenomenon without thermal fluctuations. In addition, tunneling or coherence effects can strongly couple the systems and give rise to macroscopic quantum
* Fax: 071-275404; e-mail: [email protected].
behaviors. New terms and their associated concepts are constantly appearing in the literature as for instance: quantum spins, charge and spin gaps, quantum fluctuations and frustration, ¹"0 phase transitions or quantum critical points, quantum disorder and spin glasses, spin liquid, resonanting valency bonds (RVB), spincharge separation, stripes, non-Fermi liquids, Kondo insulators, etc. Due to the difficulties of obtaining wellcharacterized samples and performing measurements at very low temperatures, experiment is sadly lacking and has fallen far behind the theoretical endeavors. In this paper we focus upon two strongly correlated uranium intermetallic compounds recently fabricated in single crystal form, UNi B and URh Ge , which ex hibit unusual magnetic properties related to the presence of disorder and frustration. Their special behavior places them in the above realm of quantum magnetism and they may serve as “test-beds” for the various theoretical predictions invoking quantum disorder and frustration. Before reviewing the experimental properties of these compounds and comparing them with recent theories, let
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 9 1 5 - 6
J.A. Mydosh / Physica B 259—261 (1999) 882—886
883
Fig. 1. Possible magnetic ground states obtainable from the perspective of site disorder and frustration.
us first try to gain some perspective regarding heavyfermion magnetism with respect to site disorder and magnetic frustration. Consider the 2;2 matrix of possible magnetic ground states shown in Fig. 1 and usually applied to insulating compounds. If we now select four representative heavy-fermion systems, we can nicely fill in the various squares with different combinations (or degree) of disorder and/or frustration — all leading to unconventional magnetic behavior of the given material. URu Si — possessing no disorder or frustration, yet an unusual “small-moment” k +0.03 l magnet ism appears via a sharp phase transition below 17 K. Superconductivity then coexists with this state at about 1 K. In spite of almost fifteen years of investigation, even now there is no agreement or “smoking gun” regarding the 17 K transition. According to the bulk properties and their field dependences, the “ordering” is related to a CDW or an antiferromagnetic array of quadrupoles [2]. Yet the definitive localprobe experiment (high-resolution X-ray diffraction or NMR) is still lacking. (ii) CePd Al — possessing disorder where single crys tals exhibit a strange multi-site randomness on the Al sublattice. Magnetic frustration is not present, however, the disorder prevents the expected longrange antiferromagnetic state. In polycrystalline samples the randomness is surprisingly less and the antiferromagnetic state does indeed form [3,4].
Fig. 2. (a) Crystallographic CeCo B-type subcell of UNi B. (b) Magnetic structure of hexagonal UNi B projected onto the basal plane. The magnetic layers are stacked ferromagnetically along the c-direction. The thin solid line represents the magnetic unit-cell. Note the presence of the “frustrated” U-spins indicated by (1) and (2) with two inequivalent magnetic environments. The nn and nnn magnetic exchanges are indicated by J and J . From Mentink et al. [8].
(i)
Due to lack of space we shall not discuss these interesting systems any further (for additional discussion see Ref. [5]) and move onto the other two compounds in Fig. 1, UNi B and URh Ge , and their quantum-magnetism behaviors.
2. UNi4B This material possesses only geometric frustration because of its hexagonal lattice and in-plane antiferromag-
netic interactions. As far as our metallurgical, and X-ray and neutron-diffraction investigations [6,7] can tell, there is no site disorder in this single-crystal compound, probably because of large size differences in the three atomic constituents. Magnetically a sharp phase transition occurs at ¹ "20 K which represents a partial anti, ferromagnetic ordering of the U-moments. Fig. 2 depicts the results of magnetic neutron-scattering experiments [8]. The most probable spin structure is sketched for the ordered U-moments (k "1.2 l at ¹"0) which are confined to the hexagonal basal plane. Note that only 2/3 of the U-ions participate in the long-range order. The remaining 1/3 are free (paramagnetic) or frustrated (spin liquid) or compensated (Kondo) spins that dominate the bulk properties at low temperatures (¹;¹ ). The long, range ordered structure is most unusual for a 2D triangular lattice (all the spins are simply ferromagnetically coupled along the c-axis). Two types of alignments surround two different sites of frustrated spin, labeled (1) and (2) in Fig. 2. But how can this occur if all the U-sites are equivalent as drawn in the figure? Answer: they are not. There is a small superstructure distortion of the crystal lattice that has been observed via X-ray and neutron diffraction. These measurements show that the surrounding U (along with the Ni and B) atoms to (1) and (2) displace in such a manner so as to slightly modify their local symmetry leaving that at U-sites (1) and (2) unchanged. Thus, these sites are “marked” by the crystallographic superstructure and this designates them as the
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centers of the unique vortex-like arrangement at (1) and the encircling three pairs at (2). For a complete description of both the magnetic and lattice structures, see Ref. [6,7]. With respect to a quantum behavior or phase transitions, the question remains as to what happens to these free, frustrated or compensated spins when the UNi B is cooled to ¹P0. Fig. 3 exhibits the low-temperature specific heat (plotted as C/¹) and the resistivity (o and do/d¹) [9]. Clearly there is a subtle transformation taking place at 0.3 K. What is exceptional here is that the c-value (,C/¹) is much more sensitive to the transformation than o(¹) or do/d¹. The latter only changes by a few percent with a broad maximum in its derivative, see Fig. 3. However, the entropy under the C/¹ versus ¹ curve to 1 K is much smaller than that if the remaining 1/3 U moments ordered. Additional measurements (not shown) of the AC-susceptibility and lSR seem to indicate no alteration in the magnetic state, i.e., there is neither a sharp magnetic response nor a rapid growth of internal fields around 0.3 K [10]. At this time we can only speculate as to the cause of this low-temperature anomaly. The behavior seems to represent a crossover either to a reorientation of the ordered U-moments spin structure, or more likely, to a new frozen quantum state of the frustrated spins. An interpretation in terms of the Kondo effect appears improbable because c does not become constant in the low ¹ limit (¹;¹ ) but instead ) traverses a maximum and continues to decrease. Furthermore, the resistivity has a clear positive slope in this ¹-regime. Two recent theoretical models have been applied to UNi B. The first by Lacroix et al. [11] treats its magnetic properties in terms of first and second-nearest-neighbor exchanges, J and J , and a Kondo-effect parameter D. By forming an effective Hamiltonian using these three parameters and minimizing its energy, the magnetic phase diagram can be obtained in parameter space. For reasonable values of J , J and D, the magnetic structure, as shown from neutron diffraction in Fig. 2, appears in the calculation. The Kondo effect is reasoned to play a key role by selectively compensating the 1/3 U-magnetic moments at sites (1) and (2). Without spin these moments cannot enter into the ordered structure and remain independent of it. However, in the process of being compensated these U-spins should act as Kondo impurities exhibiting the behavior of such, e.g., a negative temperature coefficient of resistivity, an enhanced and constant c-value at low ¹, and a Curie—Weiss-like susceptibility which also becomes constant at low ¹. Preliminary measurements given in Fig. 3, and the susceptibility, which exhibits a small broad maximum around 0.3 K, are inconsistent with a simple Kondo compensation of the frustrated U-moments. The second theory by Tejima and Oguchi [12] uses a Heisenberg Hamiltonian with various pairs of nn and
Fig. 3. Low-temperature specific heat of UNi B plotted as C/¹ versus ¹. Inset: resistivity o and do/d¹ in the same ¹-region. From Movshovich et al. [9].
nnn interacting spins in the X-½ plane; also involved is an interhexagon exchange coupling. Minimizing the (classical) energy of the Hamiltonian determines the spin configuration of the classical ground state. Indeed for proper choice of exchange parameters a spin configuring agreeing with the experimental one in Fig. 2 can be obtained, if quantum fluctuations are included to unstabilize the standard 120° (3 sublattice) Ne´el order. These fluctuations, therefore, lead to 1/3 of the U-moments remaining paramagnetic. Tejima and Oguchi have also calculated the low ¹-dependence of the specific heat. Their results are that spin waves give the dominant contribution and that C/¹ reaches a broad maximum at +2 K and smoothly decreases to zero below. This behavior is not in agreement with experiment. And so both available theories do not seem to confront or describe the unusual low ¹ behavior of UNi B. 3. URh2Ge2 Here we consider a situation where both lattice disorder and magnetic frustration are present. But how can a perfectly stoichiometric, single crystal intermetallic compound be disordered? We have already mentioned the subtle multi-sites of the Al-atoms in CePd Al that destroy the long-range antiferromagnetic order. However, for URh Ge the randomness is more dramatic and because of the randomly oscillating exchange coupling (RKKY), magnetic frustration results and a clear spinglass transition occurs. Now what is the exact cause of the disorder? Based upon extensive metallurgical, and neutron and X-ray diffraction measurements [13] we conclude: (i) there is a mixture of the two possible bct-crystal structures, ThCr Si and CaBe Ge , (ii) a random exchange of Rh and Ge sites occurs, (iii) a distribution of the free positional z-parameter of Rh and Ge takes place, and
J.A. Mydosh / Physica B 259—261 (1999) 882—886
(iv) a small amount of vacancies exists on these ligand sites. Note the magnetic U-sites are perfectly ordered, only the local environments are affected by the Rh and Ge randomness. When the above is combined with predominantly ferromagnetic interactions in the basal plane and antiferromagnetic couplings along the c-axis, we have the competing exchanges and necessary frustration to create a 3D, Ising-like, random-bond, metallic, ‘‘heavy-fermion’’ spin glass — the first of its kind [14]. Fig. 4 shows the real and imaginary parts of the linear AC-susceptibility at different frequencies u. Note the sharp u-dependent cusp of s around 10 K that shifts to higher temperatures as the frequency is increased. Also remarkable is that s for B (the driving AC field) parallel to the c-axis is much larger than s for B#aˆ . This indicates an Ising character. s appears as a much smaller step whose point of inflection corresponds to the cusp in s , and accordingly, designates the spin-glass freezing temperature ¹ (u). The frequency shift of ¹ , *¹ /(¹ *log u) is 0.025, a value typical for metallic spin glass. The DC-susceptibility, s "M/H, (not shown) also exhibits "! the characteristic behavior of an archetypal spin glass, namely, after zero-field cooling, a cusp is formed in s at "! ¹ , while upon field cooling a plateau develops below ¹ . There are further the usual irreversibilities and logarithmic time dependences when the external field is modified at ¹(¹ . The magnetic specific heat C resembles that of a canonical spin glass. C is featureless below 10 K, and a broad maximum appears in C/¹ above. At low ¹, C varies between ¹ and ¹ with the linear ¹ coefficient, c, having a value of 130 mJ/mol K in the limit ¹P0. The above data, when taken collectively, demonstrate that URh Ge is a strongly correlated spin glass.
Fig. 4. Frequency dependence of the AC-susceptibility s and s for URh Ge with the driving field parallel to the a- and c-axis. The frequencies are 1.16, 11.6, 116 and 1160 Hz. From Su¨llow et al. [14].
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Only the electrical resistivity does not show the conventional properties of metallic spin glasses [15]. Fig. 5 plots the total resistivity o(¹) up to room temperature. Here there are (i) unusually large values of o exceeding the Ioffe—Regel criterion of +200 lX cm, (ii) a large and ¹-dependent anisotropy between the a- and c-axis, and (iii) negative temperature coefficients (do/d¹(0) for both crystallographic directions. The solid line in Fig. 5 represents a fit to (p #m(¹)\ where p O0 is the ¹"0 metallic conductivity and “m” a fit parameter. The (¹-temperature dependence, which has often been observed in metallic glasses, comes from “incipient localization” or corrections to the Boltzmann formula when the inelastic mean free path is much larger than the elastic mean free path for diffusive electronic transport in a disordered medium. Along the a-axis the above fit is quite good so there is little influence of spin or magnetic scattering. However, along the c-axis clearly an additional scattering contribution is found. If we separate the data into two components, viz., disorder and magnetic, and let the dashed line in Fig. 5 represent the former contribution, then a reasonable approximation for the magnetic part is gained by subtracting the dashed line from the experimental data. The difference is o (¹), #cˆ , which
exhibits a marked drop at about 10 K (+¹ ) and be comes constant (as it should) for ¹*60 K. Hence, we have developed a method to track ¹ via the resistivity along the Ising axis. What makes this system even more intriguing is its low freezing temperature and Ising character, and thereby, its proximity to a theoretically predicted quantum critical point. Recent theories (see for example Ref. [16]) have calculated a unique temperature versus quantum fluctuation strength, r, phase diagram for a spin glass — see Fig. 6. As ¹ is lowered by increasing r, a quantum critical regime is accessed whereby a non-Fermi liquid results (II) with unusual experimental behaviors of C/¹, s, o, 1/¹ , etc. The quantum critical point govern ing these finite ¹ properties exists at r"r and ¹ "0.
Fig. 5. Resistivity versus temperature for URh Ge along the a and c axes. See text for line fits. From Su¨llow et al. [15].
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type or in tiny amounts, can greatly modify the magnetic character especially if frustration also exists. At present it is extremely difficult to characterize the disorder using metallurgical technique and what little we know is usually based on indirect physical measurements. Finally, there remain vast open areas of U intermetallic compounds where new and unexpected phenomena can be found via various combinations of elements and structures. Fig. 6. Theoretical phase diagram for a metallic spin glass. For the case of URh Ge “r”, the tuning parameter, can be disorder, pressure or transverse (to Ising axis) magnetic field. See text for description of various regions. From Sachdev and Read [16].
Increasing r beyond r we reach a quantum-disordered region (I) with its own new and distinct characteristics different from the preceding regime (quantum critical). In the experimental situation for URh Ge the quantum fluctuation parameter r may be tuned by varying the disorder via the sample preparation, applying hydrostatic pressure or using an external transverse (to the Ising axis) magnetic field. Certainly one of these three should reduce ¹ P0 and make the region around the quantum critical point (see Fig. 6) available for measurement. This would also allow entrance to the proposed quantum Griffiths singularities [17,18] where strong divergences in the susceptibilities are expected. In addition, as we know prolong annealing the single-crystal samples removes the crystallographic disorder and a long-range ordered antiferromagnetic state appears at ¹ "15 K. Thus, we have , an extremely favorable temperature region (9 to 15 K) with which to study the classical Griffiths phase by its relaxation behavior of the magnetization. URh Ge is a very promising material for future experimentation and a number of the above phenomena are presently being explored. In conclusion, we have shown that disorder and frustration play a major role in determining the behavior of strong correlated material and lead to novel effects of quantum magnetism. Disorder, even of an unintentional
Acknowledgements I wish to acknowledge my long-standing collaborations with A.A. Menovsky and G.J. Nieuwenhuys and my former Ph.D. students S.A.M. Mentink and S. Su¨llow, who performed the major work on UNi B and URh Ge . References [1] J. Phys.: Condens. Matter 8 (1996) 9675—10148, Physica B 230—232 (1997) 1—1082. [2] P. Santini, Phys. Rev. B 57 (1998) 5191. [3] S.A.M. Mentink et al., Phys. Rev. B 49 (1994) 15759. [4] H. Tou et al., J. Phys. Soc. Japan 63 (1994) 4176. [5] J.A. Mydosh, Z. Phys. B 103 (1997) 251. [6] S.A.M. Mentink, Ph.D. Thesis, Leiden University, 1994. [7] A. Drost, Ph.D. Thesis, Leiden University, 1995. [8] S.A.M. Mentink et al., Phys. Rev. Lett. 73 (1994) 1031. [9] R. Movshovich et al., to be published. [10] M.W. Meisel, G.J. Nieuwenhuys, private communications. [11] C. Lacroix et al., Phys. Rev. Lett. 77 (1996) 526. Physica B 230—232 (1997) 529. [12] S. Tejima, A. Oguchi, J. Phys. Soc. Japan 66 (1997) 3611. [13] S. Su¨llow, Ph.D. Thesis, Leiden University, 1996. [14] S. Su¨llow et al., Phys. Rev. Lett. 78 (1997) 384. [15] S. Su¨llow et al., to be published. [16] S. Sachdev, N. Read, J. Phys.: Condens. Matter 8 (1996) 9723. [17] H. Rieger, A.P. Young, Phys. Rev. B 54 (1996) 3328. [18] M. Guo et al., Phys. Rev. 54 (1996) 3336.