Anomalous metamagnetism in the low carrier density Kondo lattice YbRh3Si7

We report complex metamagnetic transitions in single crystals of the new low carrier Kondo antiferromagnet YbRh3Si7. Electrical transport, magnetization, and specific heat measurements reveal antiferromagnetic order at T_N = 7.5 K. Neutron diffraction measurements show that the magnetic ground state of YbRh3Si7 is a collinear antiferromagnet where the moments are aligned in the ab plane. With such an ordered state, no metamagnetic transitions are expected when a magnetic field is applied along the c axis. It is therefore surprising that high field magnetization, torque, and resistivity measurements with H||c reveal two metamagnetic transitions at mu_0H_1 = 6.7 T and mu_0H_2 = 21 T. When the field is tilted away from the c axis, towards the ab plane, both metamagnetic transitions are shifted to higher fields. The first metamagnetic transition leads to an abrupt increase in the electrical resistivity, while the second transition is accompanied by a dramatic reduction in the electrical resistivity. Thus, the magnetic and electronic degrees of freedom in YbRh3Si7 are strongly coupled. We discuss the origin of the anomalous metamagnetism and conclude that it is related to competition between crystal electric field anisotropy and anisotropic exchange interactions.


I. INTRODUCTION
Materials containing partially-filled f orbitals are of great interest to the strongly correlated electron system community because of their quantum complexity. This is driven by several competing parameters that include Kondo coupling (which favors a non-magnetic ground state with enhanced effective mass), Rudermann-Kittel-Kasuya-Yosida interactions (RKKY, which favors long-range magnetic order), and crystal electric field (CEF) effects (which left the degeneracy of the Hund's rule ground state multiplet). Among f -electron systems, the ground states of many Ce-, Yb-and U-based compounds are highly susceptible to tuning by nonthermal control parameters, such as pressure, chemical substitution, or magnetic field 1 . This often results in emergent phenomena such as unconventional superconductivity 2-4 , non-Fermi liquid behavior near a quantum critical point 3,5-9 , hidden order 4 and metamagnetism [10][11][12][13][14][15][16][17][18] .
In several materials, metamagnetism has been linked with magnetic quantum criticality 16,19 , although the origin is unclear.
Interestingly, very few Yb-based metamagnetic (MM) compounds have been reported to date 20,[22][23][24] , compared to a larger number of Ce-based 13,14,17,18,25? -28 and U-based 10,11 MM compounds. Here, we report the discovery of metamagnetism in single crystals of the Kondo lattice YbRh 3 Si 7 . YbRh 3 Si 7 is the first compound displaying either Kondo correlations or metamagnetism in the ScRh 3 Si 7 (1-3-7) family 29,30 , for which the only known rare earth-based systems are non-magnetic RAu 3 Al 7 (R = Ce-Sm, Gd-Lu) 31 , magnetic Eu(Rh,Ir) 3 Ge 7 32 , and YbAu 3 Ga 7 with unknown physical properties 33 . In YbRh 3 Si 7 , the Kondo effect is clearly indicated by Kondo lattice-like resistivity, reduced magnetic entropy released at T N , and density functional theory (DFT) calculations. Anisotropic magnetic susceptibility and specific heat measurements reveal a long-range magnetic ordering transition at T N =7.5 K. Neutron diffraction measurements confirm that the zero field ordered state is antiferromagnetic (AFM), with the moments lying in the ab plane.
We present high-field magnetization, torque, and resistivity up to 35 T. Around 2 K, these measurements reveal two field-induced MM transitions at µ 0 H 1 = 6.7 T and µ 0 H 2 = 21 T along the c axis. Angular dependent magnetoresistivity shows that both H 1 and H 2 increase monotonically when the crystal is rotated away from the c axis towards the ab plane.
When H ⊥ c, only one MM transition is found, up to the maximal field, at µ 0 H 1 = 10 T.
This behavior is starkly different from what has been observed in other MM materials; MM transitions are rarely observed for a field orthogonal to the moments, and, if present, typically occur at higher fields than those for the field parallel to the moments. Given that the easy axis, determined by the CEF anisotropy, is along the c axis, while the ordered moment is lying in the ab plane, the anomalous metamagnetism in YbRh 3 Si 7 may be a result of the delicate balance among different underlying energy scales, including CEF anisotropy and exchange anisotropy. Understanding the metamagnetism in YbRh 3 Si 7 will help draw a more complete picture of how subtle quantum effects steer the macroscopic behavior of different materials.   drop below T * is strongly field dependent ( Fig. 1(b)), indicating a magnetic phase transition.

Single crystals of YbRh
Magnetization, neutron diffraction, and specific heat measurements, which will be discussed below, show that YbRh 3 Si 7 has an AFM ground state with T N = 7.5 K. This is relatively high among Yb-based antiferromagnets, with very few other such systems showing comparable ordering temperatures (Yb 3 Cu 4 Ge 4 with T N = 7.5 K 46 , YbRhGe with T N = 7 K 47 , and Yb 2 MgSi 2 with T N = 9.5 K 48 ).
Next we turn to the field dependent properties of YbRh 3 Si 7 , starting with the electrical resistivity and Hall effect. Large positive magnetoresistance at 2 K is apparent from the data measured at µ 0 H = 0 and 9 T (full and open symbols, Fig. 1(b)). The Hall coefficient R H ( Fig. 1(c)) is positive and strongly temperature-dependent. Within a single-band picture, the carrier density n = 1/eR H is estimated to be 1.5 × 10 21 cm −3 at 2 K, one to two orders of the magnitude smaller than in a regular metal. We can therefore conclude that YbRh 3 Si 7 is a low carrier Kondo lattice antiferromagnet.
The H = 0 magnetic ordering transition is confirmed by anisotropic magnetization measurements ( Fig. 1(d)). When the magnetic field is applied parallel or perpendicular to To shed light on the magnetic properties of YbRh 3 Si 7 , electronic band structure calculations were performed using the DFT and DFT+U techniques, as described in the METHODS section. We have found the lowest energy configuration to be the one in which the Yb magnetic moments are ordered ferromagnetically within the ab plane, pointing along the c axis, with AFM order between adjacent planes. Next, we present the partial DOS projected onto different orbital angular momentum components, m L of the Yb 3+ ion, plotted in Fig. 1(e).
Top (bottom) panels show the minority (majority) spins, accordingly. Considering any given Yb ion, the minority spin DOS is dominated by m L = 2 (red) at the Fermi level (vertical dotted line). The other m L orbitals, which are represented by the blue lines, lie below the Fermi level and hence do not contribute to the moment. This corresponds to the spin-orbit coupled state |J = 7/2, m J = 5/2 , as shown by the expression in the inset in Fig. 1(e). We conclude that the ground state doublet is thus |J = 7/2, m J = 5/2 , and expect that, under large magnetic fields, this state will become fully polarized. Accordingly, the calculated saturated moment µ calc sat = (L z + 2S z )µ B / should be 2.86 µ B . We next present neutron diffraction measurements, which allowed us to identify the zero field magnetic ground state of YbRh 3 Si 7 . Upon cooling below T N = 7.5 K, these measurements reveal the formation of additional Bragg reflections as shown in Fig. 1(f).
These magnetic Bragg reflections were indexed with a k = 0 propagation vector in the R3c space group. The best agreement with the measured diffraction pattern was obtained with the Γ 5 irreducible representation, which can be written as a linear combination of two basis vectors (3 · ψ 5 + ψ 6 ) 38 . The resulting refinement is shown in Fig. 1(f). In this collinear AFM structure, the spins are constrained to lie in the ab plane, as represented for a single unit cell in the inset of Fig. 1(f). The ferromagnetically-ordered planes are stacked antiparallel along the c axis. This magnetic structure has also been verified with single crystal neutron diffraction measurements. Figure 1(g) presents the agreement between the measured and calculated structure factors. The temperature dependence of the (0,2,-5) magnetic Bragg peak, shown in Fig. 1(h), reveals that at 4.5 K the intensity has not saturated. Thus, it is not surprising that the ordered moment determined from single crystal diffraction, 0.36 µ B /Yb 3+ at 4.5 K, is slightly smaller than the moment derived from powder diffraction, 0.47 µ B /Yb 3+ at 1.5 K. This partial order parameter gives T N around 7.5 K, in agreement with the magnetic susceptibility measurements in the inset of Fig. 1(d). It is worth highlighting that the structure determined by neutron diffraction differs from the one used in the DFT calculations; while both structures have alternating AFM coupled planes, in DFT the moments point along the c-axis whereas experimentally they are found to point along the a-axis. This discrepancy is likely due to the fact that DFT may not properly account for the crystal electric field anisotropy.

IV. KONDO EFFECT AND HYBRIDIZATION
With the magnetic ground state resolved from neutron diffraction measurements, a better characterization of the correlations in YbRh 3 Si 7 is needed, since the H = 0 resistivity, shown in Fig. 1(a), hinted at possible strong correlations and Kondo screening below ∼ 30 K. Long range AFM order at T N = 7.5 K is marked by a broad peak in specific heat C p (Fig. 2(a)), consistent with the magnetic susceptibility and neutron data. For comparison, the specific heat data of the non-magnetic analogue LuRh 3 Si 7 (solid line) is also shown in Fig. 2(a). The log-log plot of C p /T vs. T in the inset reveals a power-law divergence of C p /T for YbRh 3 Si 7 between 0.7 and 0.05 K. In this temperature range, such an increase of C p /T on cooling is often associated with an energy splitting of the nuclear quadrupole states in an electric field gradient of the individual atomic environment, i.e., C p /T ∼ 1/T 3 . However, as shown in the inset of Fig. 2(a), the data better described by a faster power law increase C p /T ∼ 1/T 2.5 .
This might indicate that the enhanced low T specific heat is a convolution of a nuclear Schottky contribution and enhanced electronic specific heat contribution due to hybridization of the f and conduction electron bands. A lower bound estimate of the magnetic entropy is obtained if we disregard the divergent contribution at the lowest temperatures. This lower bound is determined by assuming that below 0.7 K, C p /T is constant at 0.018 J/mol K 2 (red dashed line in inset of Fig. 2(a)). Then we subtract the non-magnetic contribution as approximated by LuRh 3 Si 7 . The resulting magnetic entropy of YbRh 3 Si 7 amounts to only 1/3 Rln2 at T N (Fig. 2(b)), implying strong Kondo correlations, with a Kondo temperature around 15 K determined from S mag (0.5 T K ) = 0.4 Rln2. Even if we include the full divergent contribution at low temperatures, our entropy estimate only increases by a very small amount.
Thus, the S mag in Fig. 2(b) can be viewed as slightly lower than the intrinsic value for YbRh 3 Si 7 , and a more accurate determination will rely on a precise estimate of the nuclear Schottky term, for which Mössbauer and NMR measurements are necessary.
To elucidate the nature of the hybridization between Yb f and conduction electron bands inferred from the thermodynamic measurements above, DFT+U calculations were performed 42 inside the AFM phase. The representative band structure is shown in Fig. 2 along the high-symmetry lines in the Brillouin zone. We have separated the partial contribution of the Yb f electrons (Fig. 2(c)) from that of the conduction electrons of Rh and Si (Fig. 2(d)) using the "fat band" representation, such that the thicker bands denote the larger contribution of the respective atomic orbitals. The results paint the canonical picture of hybridization between the very thin (atomic-like) Yb f -band and the parabolic conduction band, the latter with a bandwidth on the order of 1.5 eV. The chemical potential, pinned to the Yb f level, crosses the conduction band along the Γ-Z direction, while the spectrum remains gapless (metallic in character) even in the AFM phase. The same observation is true for the paramagnetic (PM) phase (see Fig. S3 in Supplementary Materials B).
An additional outcome from the band structure calculations is an estimate of the carrier density n ∼ 3.2 × 10 21 cm −3 in the PM phase, and 2.9 × 10 21 cm −3 in the AFM phase.
These values are close to the experimental one, n exp ∼ 1.5 × 10 21 cm −3 inferred from the low-temperature Hall coefficient (Fig. 1(c)). This comparison is favorable, especially given that the experimental value was obtained within a simplified single-band model. Also, the DFT and DFT+U calculations are both based on a single-particle picture and, as such, do not capture the many-body Kondo-lattice phenomena. Nevertheless, the band structure calculations provide a reasonable starting point for the periodic Anderson model, and predict the system to be a low-carrier density Kondo metal, in agreement with the experimental results.

V. HARD AXIS METAMAGNETISM
Having provided evidence for the low carrier Kondo metal character in YbRh 3 Si 7 , we now focus on how these properties are intertwined with the even more complex magneto-transport properties when the magnetic field is orthogonal to the H = 0 moment direction. In the H c field-dependent specific heat, T N appears to be suppressed slightly by fields up to ∼ 6 T (inset Fig. 3(a)). At higher fields, an additional transition becomes visible, and is marked by an arrow in Fig. 3(a) for µ 0 H = 6.15 T. As the field is further increased, the peak associated with this new transition becomes larger, sharper, and shifts to the higher temperature, reaching 6.8 K at µ 0 H = 9 T. In contrast, for H ab (Fig. 3(b)), no additional transition is observed up to 9 T and and the H = 0 peak in C p monotonically shifts to higher temperatures. The behavior of C p with H ab is reminiscent of the "Vollhardt invariance" that is associated with the transfer of specific heat weight under the constraint of magnetic entropy conservation. This effect is attributed to spin fluctuations above the ordering temperatures 51 . Similar behavior has been reported in several other strongly correlated systems, such as CeCu 5.5 Au 0.5 , MnSi, and CeAuSn 52-54 .
The magnetic susceptibility (Fig. 1(d)) and specific heat data (Fig. 3(a) and (b)) point to complex field-induced magnetic transitions and large CEF anisotropy in YbRh 3 Si 7 . Fielddependent thermodynamic and transport property measurements allow for an in-depth characterization of this complex magnetism. Low temperature magnetization measurements M (H) up to 7 T not only confirm the magnetic anisotropy, but revealed a MM transition above µ 0 H = 6 T for H c (solid line, Fig. 3(c)). In the orthogonal direction, no MM transition is observed (dashed line, Fig. 3(c)). The low field M ab shows a sharp increase with H and a small hysteresis at very small fields (inset of Fig. 3(c)). Due to the equivalence of the pinning picture because the spin configuration within the ab plane is not unique. Indeed, this magnetic structure is composed of three symmetrically-equivalent domains, which can be generated from the configuration shown in Fig. 1(f) by successively rotating the spins by 120°in the ab plane. Once a field is applied within the ab plane, this symmetry is broken and the domain most closely aligned with the field will become energetically most favorable, resulting in its selection to form a single domain state. The very small ∼ 0.01 T barrier to this selection and the ∼ 0.002 T hysteresis are the result of domains being pinned by the low level of disorder that is present in any material.
To investigate the evolution of M c and temperature dependence of the MM transition at higher fields, we performed magnetization, torque, and magnetoresistance measurements up to 35 T as shown in Fig. 3(d-g). All measurements were carried out for both increasing and decreasing fields, and they indicate two MM transitions, around µ 0 H 1 ∼ 6.7 T and µ 0 H 2 ∼ 21 T. The values of µ 0 H 1 and µ 0 H 2 were determined from the peaks or dips in the derivatives, as shown in the right axes of Fig. 3 (e-g). The magnetization (Fig. 3(e)) varies linearly with H up to the first critical field µ 0 H 1 . Following the MM transition at µ 0 H 1 , a linear increase of M c is seen for fields between 10 T and 21 T, followed by the second MM transition around µ 0 H 2 = 21 T. At higher fields, the magnetization approaches a plateau close to 2.8 µ B /Yb, much less than the Hund's rule ground state value of 4.5 µ B for Yb 3+ . However, the measured value is consistent with the expected saturated magnetization for m j = 5/2, which was predicted by DFT calculations to be the CEF ground state (Fig. 1(e)). The critical fields µ 0 H 1 and µ 0 H 2 are also indicated by a change of slope in the magnetoresistance and torque data as shown in Fig. 3(d,f-g). Temperature dependent magnetoresistance isotherms for H c point to an increasing critical field value for µ 0 H 1 and a decreasing critical field value for µ 0 H 2 with increasing temperatures (Fig. 3(d)). The two transitions converge around 12 T for T ∼ 8 K. When the applied field is larger than 1.6 T, the CEF easy axis anisotropy dominates and no crossover is expected, in contrast to the low field data shown in the inset of Fig. 1(d).
This is likely the result of field tuning of direct competition between the single ion CEF anisotropy and anisotropic exchange interactions, with the former dictating a magnetic easy axis parallel to the c axis and the latter leading to the low field moment alignment in the ab plane. However, this competition alone cannot explain the abrupt increase in resistivity that occurs at the onset of the MM transition at µ 0 H 1 (Fig 3(g)). In addition, in angle dependent magnetoresistivity measurements (Fig. S5), as the magnetic field is rotated away from the c axis, a monotonic increase of µ 0 H 1 and a monotonic decrease of resistance are observed, instead of extremes around θ = 45 degrees, i.e., the lowest energy state in the CAFM phase.
There are some other possible underlying energy scales that are not discussed in this study including (1) a change in the Yb moment with field due to an excited CEF level crossing the ground state, such as a low-lying state below 1 meV, (2) a structural transition with field, or In summary, YbRh 3 Si 7 is a new low-carrier, antiferromagnetic, Kondo-lattice compound with anomalous metamagnetism. It serves as a forerunner among Ce-and Yb-based 1-3-7 analogues, rendering the 1-3-7 structure an ideal host structure to investigate the intertwinement of multiple energy scales including RKKY, Kondo, CEF anisotropy, and anisotropic exchange interactions.

Supplementary Materials
A. Crystallographic analysis Table S1 shows the crystallographic data for as-grown and 150 hours annealed samples, with the atomic parameters of the latter given in Table S2.  0.0034(5) 1 a U eq is defined as one-third of the trace of the orthogonalized U ij tensor.

B. Details of the band theory calculations
In the main text, we showed that the DFT+U calculations in the AFM phase predict the |J = 7/2, m J = 5/2 state as the ground state doublet of YbRh 3 Si 7 . For completeness, here we show the results obtained without including the Hubbard U interaction. The projected DOS plot in Fig. S2 shows that the minority m L = 2 orbital (red line) has not fully split off from the other orbitals, unlike in the DFT+U case ( Fig. 1(e)). In this case, we find that the ordered moment is 0.75 µ B /Yb. The inclusion of Hubbard interaction of U = 4 eV on the Yb site further increases the moment to 1.5 µ B /Yb, a characteristic of the DFT+U framework.
Sufficient Hubbard interactions also cause the m L = 2 orbital to split from the other orbitals, as shown in Fig. 1.
We note that both the DFT and DFT+U values of the ordered moment exceed the experimental value 0.36 µ B /Yb obtained from the single crystal neutron diffraction. We attribute this disparity to the fact that the DFT-based single-particle framework cannot capture the many-body effects of the Kondo interaction and thus overestimates the bandwidths of Yb f -orbitals (and underestimates the effective mass). Additionally, the ordered moments are highly sensitive to the position of the Fermi level within the f -band. Despite this deficiency, the DFT+U calculations nevertheless provide an adequate qualitative explanation of the magnetic properties, and in particular correctly predict the AFM structure and the the nature of the ground state doublet, together with the associated saturated moment (see Further information about the electronic properties in the PM state is obtained from the DFT band structure, shown in Fig. S3 where we have separated the partial contribution of Yb f -electrons ( Fig. S3(a)) from that of the conduction electrons of Rh and Si (Fig. S3(b)).
In this so-called "fat band" representation, the thicker bands denote the larger contribution of the respective atomic orbitals. This figure is to be compared with Fig. 2(c-d) in the main text, the latter computed in the AFM phase using the DFT+U method. The two results are qualitatively the same in that they both show the hybridization between the Yb f -electrons and the conduction bands of Rh and Si. The difference is that in the present PM case, all 4 Kramers-degenerate f -bands corresponding to J = 7/2 states appear close to the Fermi level and hybridize with the conduction electrons, whereas in the AFM case, the DFT+U calculation correctly selects out the m j = ±5/2 state which appears at the Fermi level, as described above.