High-resolution X-ray diffraction

We show in Fig. 1 representative scans of both the structural and magnetic order at high pressure and low temperature, P=16.1 GPa and T=5.2 K, exemplifying the high-resolution X-ray magnetic diffraction technique. GdSi has a collinear spin structure between the Néel temperature, T N =54.5 K, and the spin-flip transition temperature, T SF =53.0 K, and an orthorhombic lattice structure. Below T SF at ambient pressure, the spin structure is planar and is associated with a monoclinic lattice structure (space group P2 1 /c) that can be detected by a split of the d-spacing between (0, K, L) and (0, K, −L)10. There are then two magnetic domains with Q vectors along (0, ±q b , q c ) associated with the individual monoclinic lattice domains (see Fig. 1d and Fig. 2b, inset). At 16.1 GPa, the highest pressure that antiferromagnetism was directly observed, the monoclinic splitting of the (0, 1, 1) order remains clear (Fig. 1f), although the longitudinal line shapes are slightly broadened above the instrument resolution21. Here, both the spontaneously twinned single crystals and near-resolution-limited diffraction profiles testify to the quasi-hydrostatic pressure condition in the sample environment. At 16.1 GPa, the full widths at half maximum (FWHMs) of the peaks give a coherence length of 1,400 Å for both the lattice and the antiferromagnetic order.

Figure 2: Pressure dependence of both lattice structure and magnetic order parameter. (a) Lattice constants, a, b, c, and the monoclinic angle α as a function of pressure. Below P c =16.4 GPa, lattice constants a, b, c are fit with the single-parameter Birch equation of state specified in the text. α(P) is fit linearly. Above P c , there is a first-order structural transition to an orthorhombic phase. (b) The magnetic order parameter I mag /I (020) is summed over diffraction intensities at (0, 2−q b , ±q c ) and (0, q b , ±q c ) for both magnetic domains. (Inset) Magnetic domain distribution at each pressure point where the order parameter was measured. Error bars represent 1σ s.d. Full size image

We plot in Fig. 2a the substantial change in the lattice constants with P. Each refined lattice constant, a, b and c, is fit with the single-parameter Birch equation of state: P=3/2 B l [(l 0 /l)7−(l 0 /l)5], where l 0 is the individual lattice constant at ambient pressure. The three bulk moduli, B a =83.1±0.4 GPa, B b =88.0±0.3 GPa and B c =85.4±0.4 GPa, are close in value, indicating that the crystal structure evolves in a nearly isotropic fashion under pressure. Overall, the antiferromagnetism in GdSi persists over a 4.7% compression of the linear lattice constants and a 13% reduction in volume. At ambient pressure, the monoclinic angle α of the lattice tracks the evolution of the planar magnetic order via magnetostrictive coupling10. Up to 16.1 GPa, α changes only slightly and deviates slowly away from 90°. At a critical pressure, P c =16.4 GPa, the a-axis lattice constant shows a clear discontinuity of Δa/a=5.9 × 10−3, while changes along the other two axes are minimal. Beyond P c , no splitting of the (0, K, L) orders is observed for three different crystals as the lattice structure again becomes orthorhombic (the structure found for T>T SF at ambient pressure). The incommensurate antiferromagnetic order is no longer visible beyond P c , and the nature of the high-pressure phase, whether it is a paramagnet, a ferromagnet or a commensurate antiferromagnet, cannot be discerned from this data.

Antiferromagnetic ground state under pressure

The incommensurate antiferromagnetism in GdSi and its evolution with P can be probed directly by observing the magnetic order at points in reciprocal space such as (0, q b , ±q c ), (0, 2−q b , ±q c ) and (0, 2+q b , ±q c ). The magnetic nature of these incommensurate diffraction peaks was demonstrated previously at ambient pressure via resonant X-ray diffraction, and no lattice (charge) coupling at this wave vector was detected10. As shown in Fig. 2b, and in telling contrast to the behaviour of the lattice, the magnetic order parameter remains unchanged over the entire range of pressures, 0≤P<P c =16.4 GPa. The magnetic order parameter was measured by summing over the diffraction intensities of both domains (Fig. 2b, inset). In the diffraction geometry of a diamond anvil cell, it is difficult to measure a full azimuthal dependence under pressure10. However, given that the data were taken at a fixed azimuthal angle (see Methods), both the constant diffraction intensity and lattice monoclinic angle substantiate the interpretation that the spin structure is invariant under pressure.

In addition to the strength of the magnetic order parameter, the stability of the incommensurate spin structure in GdSi under pressure is further illustrated through measurements of the wave vector Q. We show in Fig. 3, Q(P) at fixed T and Q(T) at fixed P at the approaches to the quantum and thermal transitions, respectively. Along the temperature path, there is a significant change in Q(T) that accelerates near the phase boundary in a manner characteristic of many incommensurate SDW and CDW systems8,22. Under pressure, however, Q(P) of GdSi stays essentially constant (relative to the lattice) as it approaches the first-order, high-pressure phase boundary, with a slight, monotonic decrease in q b (P); the germane characteristics of the Fermi surface nesting are unaffected. We note that Q(P) would behave differently for systems approaching a continuous quantum phase transition. A non-monotonic Q(P→P c ) has been measured in the incommensurate SDW in Cr arising from fluctuation effects at the approach to the quantum critical point11. Here, in GdSi, the magnetism does not weaken with applied pressure and quantum fluctuations have been cut off by the first-order structural transition.