Modulating superconductivity
Strain can have considerable effects on the electronic properties of materials. For instance, the temperature at which a material becomes superconducting—the critical temperature—can be tuned by varying strain. Bachmann et al. used focused ion beam milling to fabricate microstructures of the superconductor CeIrIn5 on sapphire substrate. A difference in the thermal contraction coefficients of the two layers induced nonuniform strain upon cooling of the sample, leading to large gradients of the critical temperature. This approach can be used in other materials and may enable fabrication of superconducting circuitry without physical junctions.
Science, this issue p. 221
Abstract
Although crystals of strongly correlated metals exhibit a diverse set of electronic ground states, few approaches exist for spatially modulating their properties. In this study, we demonstrate disorder-free control, on the micrometer scale, over the superconducting state in samples of the heavy-fermion superconductor CeIrIn5. We pattern crystals by focused ion beam milling to tailor the boundary conditions for the elastic deformation upon thermal contraction during cooling. The resulting nonuniform strain fields induce complex patterns of superconductivity, owing to the strong dependence of the transition temperature on the strength and direction of strain. These results showcase a generic approach to manipulating electronic order on micrometer length scales in strongly correlated matter without compromising the cleanliness, stoichiometry, or mean free path.
The electronic ground state of heavy fermions sensitively depends on the coupling between a localized state and an itinerant electronic system. As the coupling strength is tuned, metallic, superconducting, or magnetically ordered phases are induced, yielding the rich phase diagrams typical for materials of this class. This tunability could be exploited for device-based applications if the electronic order can be locally controlled within a crystalline sample: Metallic, magnetic, or superconducting regions could be induced within a single crystal by precise spatial control over the tuning parameter. Strain is a particularly powerful way to achieve this goal: It introduces no disorder, and its independent components offer multiple degrees of freedom to couple to electronic order. Most commonly, uniform uniaxial strain (1) or biaxial strain (2) is applied. In this work, we demonstrate micrometer-scale control over the superconducting order in stoichiometric and ultraclean CeIrIn5 by inducing a nonuniform tailored strain field in microstructured single-crystal devices. Our experimental approach exploits strain induced by differential thermal contraction between the sample and the substrate, as well as our submicrometer control over the shape of the sample. The tetragonal heavy-fermion metal CeIrIn5 exhibits material parameters that are ideal for establishing spatial control of a correlated state [Sommerfeld coefficient γ ~ 720 mJ mol−1 K−2 (3), effective mass m* ~ 30me, where me is the mass of an electron (4)]. The superconducting transition temperature, Tc, of this material is highly sensitive to strain, owing to the strong dependence of Ce 4f hybridization on the Ce-Ce interatomic distance. Straining the sample along the a direction increases the bulk superconducting transition temperature
Figure 1 illustrates nonuniform superconductivity for the simple case of a rectangular slab also referred to as a lamella. The lamella (150 μm by 30 μm by 2 μm) was carved from a macroscopic crystal by using focused ion beam (FIB) machining [for fabrication details, see (8–10)] and was joined to the sapphire substrate with a thin layer of epoxy (approximately a few hundred nanometers thick). This epoxy is substantially softer than the crystalline substrate and the sample; finite element modeling corroborates the intuitive assumption that the differential thermal contraction transmitted through the epoxy is largely independent of the exact details of the glue layer, such as its thickness or elastic moduli [see (10) for details]. The crystallographic c direction is aligned with the short side of the lamella and the a direction with the long side. The use of sapphire cut along the (0001) surface ensures isotropic thermal contraction of the substrate. Whereas sapphire is known for its low thermal contraction, CeIrIn5 contracts strongly upon cooling, as is typical of many Ce-based compounds (11). As a result, the sample is under tensile strain at low temperature.
(A) Sketch of the distortion of a thin lamella of CeIrIn5 coupled to sapphire at low temperatures. (B) Optical image of a 2-μm-thick lamella cut by FIB machining in the (a, c) plane. (C) Finite element simulation of the Tc map across the sample, arising from the strain profile and strain dependence of Tc (10). (D to F) (Top) Local susceptibility images at three representative temperatures. A negative diamagnetic susceptibility indicates superconducting regions of the sample. The susceptibility is measured in units of superconducting magnetic flux quanta (Φ0) detected in the SQUID per ampere (A) applied to the field coil. (Bottom) Superconducting regions (white) calculated from the strain profile in the device, corresponding to constant temperature contours of the Tc map in (C).
To study the superconducting transition in the lamella, we use scanning superconducting quantum interference device (SQUID) microscopy (SSM) to image the diamagnetic response of the sample with micrometer-scale resolution. To detect superconductivity, we apply a local magnetic field by sourcing a current through a ~6-μm field coil integrated on the SQUID chip. The ~1.5-μm SQUID pickup loop detects the local magnetic susceptibility measured in magnetic flux per unit current in the field coil [see (10) for details]. Superconducting regions of the sample exhibit a strong diamagnetic response, which enables us to distinguish them from metallic and insulating regions.
Susceptibility images as a function of temperature (top images in Fig. 1, D to F) reveal that superconductivity first emerges at the short edges while most of the lamella remains metallic. As the temperature is lowered, larger fractions of the lamella become superconducting, thus leading to the growth of triangular superconducting patches protruding into the lamella, which eventually join in the center (Fig. 1E). At even lower temperatures, the order parameter remains suppressed on the long edge. The observation of superconductivity at the edge of the sample before the interior is unexpected. Usually, for a thin superconducting slab cooled in Earth’s magnetic field, the demagnetization factor at the sample edges initially favors the appearance of superconductivity in the center of the sample (12).
Before analyzing the spatial pattern in the images, we estimate whether strain caused by differential thermal contraction combined with the strain sensitivity of Tc in CeIrIn5 can cause the observed variations in Tc of several hundred millikelvin. When cooled to cryogenic temperatures, CeIrIn5 and sapphire contract by ~0.3 and ~0.08%, respectively. Given this mismatch, we expect strain on the order of 0.1% to exist within the CeIrIn5 crystal at low temperature. Using a typical elastic modulus of 150 GPa, this is equivalent to a uniaxial pressure of ~1.5 kbar, which changes Tc by ~100 mK (5). Hence, both the uniaxial pressure studies and the image series presented here are consistent with ~0.1% strain generating ~100 mK variation in Tc.
To understand the patterns of superconductivity, we perform finite element method simulations of the device’s strain field, which is caused by the difference in the thermal contraction between CeIrIn5 and sapphire (10). We then compute the local transition temperature, Tc, from the strain field using
Next, we show that the induced strain field and the shape of the superconducting regions can be tailored by using FIB micromachining. To define the strain field in the devices, additional trenches were cut through the lamella down to the substrate, changing the boundary conditions for the elastic equations. In both devices shown in Fig. 2, the trenches define a square in the (a, c) plane. In device 1, the square is anchored by four constrictions, one in each corner (Fig. 2A). In device 2, the constrictions connect to the center of each side of the square (Fig. 2G). Under biaxial tension, each of the contact pads is pulled outward, subjecting the square to nonuniform strain.
(A) Scanning electron microscopy (SEM) image of device 1. The lamella is FIB-cut in the (a, c) plane and contacted by evaporated Au (yellow). The center square of the device is 25 μm by 25 μm by 3 μm, held by contacts in the corners. (B to D) (Top) Local susceptibility images at three representative temperatures illustrate the temperature evolution of the spatially modulated superconducting state. (Bottom) Calculated superconducting patterns (10). (E) Montgomery transport measurement upon cooling of device 1. As the first superconducting regions appear on the sides along the c direction (D), the c-direction resistance Rc = Vc/Ic vanishes and the a-direction resistance Ra = Va/Ia experiences a resistance spike, caused by a sudden current redistribution. I, current; V, voltage. At lower temperatures, these regions touch (C), leading to zero resistance across all contacts. (F) Tc map for device 1 from finite element calculations (10). (G) SEM image of device 2. The fabrication of device 2 was as similar as possible to that of device 1, but with the constrictions connecting at the middle of each side of the square, not the corners [compare with (A)]. (H to K) The same as in (B) to (D) and (F), but for device 2. A completely different superconducting pattern develops, as expected from the difference in position of the contacts.
Device 1 is designed to measure anisotropic resistances in the plane by passing current through any pair of neighboring contacts while measuring voltage across the remaining pair (13, 14). In this device, the simulated Tc map predicts a pattern of superconductivity first developing on the edges aligned with the c direction as the device is cooled (Fig. 2F). These regions extend toward the center upon lowering the temperature, eventually connecting in the middle of the device. Such patterns are evident in the SSM images (Fig. 2, B to D) and lead to three distinct regimes for transport through device 1: (i) the normal state in which all contacts are separated by metallic regions, (ii) a state in which only the contact pairs along the c direction are connected by superconducting regions, and (iii) a state in which all contacts are connected by a single superconducting region. As a result, when current is sourced between contacts along the c direction (1 and 2), a transition to zero voltage, signaling superconductivity at a relatively high temperature
In contrast to those of device 1, the edges in device 2 parallel to the a axis superconduct well before the central region of the device (Fig. 2, G to K). As with device 1, we find that simulations of the strain profile in the device reproduce the structure of the superconducting transition in detail. The contrast between the two image series indicates that the structure in the images is determined by the interplay between the intrinsic strain sensitivity of the material and the strain field imposed by the FIB-defined features.
In device 2, we observe a pronounced suppression and enhancement of Tc in the a- and c-aligned constrictions, respectively. The strain in the constrictions is enhanced because the contact pads on one side and the square on the other side exert forces on the constrictions that point outward. This observation suggests a strategy to design devices that exhibit a strong modulation of Tc and to generate confined regions of suppressed superconductivity.
Smaller devices exhibit an even more pronounced dependence of the transport Tc on their geometry than larger devices. Device 3 features three series-connected straight beams with dimensions 22 μm by 1.8 μm by 8 μm, with two beams aligned with the c direction and one with the a direction (Fig. 3A). These fine structures cannot be resolved in detail with SSM. In transport, the transition temperature for the c-aligned beam
(A) SEM image of device 3. The device consists of long bars with dimensions 1.8 μm by 8 μm by 22 μm. Two bars are oriented along the c direction and one along the a direction. (B) Resistivity as a function of temperature for device 3. Strain suppresses Tc along the bar aligned with the crystallographic a axis and enhances Tc along bars aligned with the c axis. (C) Temperature-dependent magnetoresistance along the a and c directions in CeIrIn5 for different field orientations. All measurements were taken in a full Lorentz force configuration except ρc for
To exploit strain tuning of the superconducting order for future device-based experiments, the superconducting state must remain robust after device fabrication. To estimate the critical current of each beam, we apply high currents to device 3 (Fig. 4, A and B). The current is increased until an observable voltage signals the breakdown of the zero-resistance state. To minimize self-heating, 83-μs rectangular current pulses are applied to the sample, with a cooldown time between pulses of 100 ms. Because the critical current decreases monotonically with increasing temperature, the obtained values represent a lower bound of their magnitude in the absence of heating. Figure 4A shows a typical current–voltage characteristic of the c-aligned beam at 500 mK, well above the bulk Tc. A robust zero-resistance state is detected up to a critical current density of ~12.5 kA cm−2. Upon cooling, the critical current density increases to
(A) Current–voltage characteristic along the c direction measured on device 3 (see Fig. 3A). The onset of measurable voltage above the experimental noise level (arrow) was used to define
Incidentally, our observations answer an open question about the origin of the commonly observed discrepancy between thermodynamic and resistive measurements of Tc in CeIrIn5. Bulk crystals display a transition to a zero-resistance state well above Tc, starting as high as
Here, we report a strategy to spatially modulate superconductivity within a clean electronic system that is induced by strong, nontrivial strain patterns. Spatial gradients of Tc have been generated by other methods—for example, by modulating the chemical potential across the sample by gradient doping with molecular-beam epitaxy techniques (23). In this method, variations in the local charge carrier density modify the local transition temperature. In the stoichiometric CeIrIn5 microstructures that we studied, the charge carrier density is uniform, as evidenced by unperturbed quantum oscillations in device 3 (Fig. 4, C to E). These quantum oscillations quantitatively match the angle dependence of previously reported de Haas–van Alphen oscillations (4) measured on macroscopic crystals and indicate that the Fermi surface shape remains unchanged by the weak strain field. In particular, the large, heavy orbits and their fine structure are readily observed, which is usually very difficult in transport. This observation is incompatible with the presence of strong charge carrier density changes across the sample, which would lead to spatial variations in the Fermi surface cross sections and subsequently suppress quantum oscillations by phase smearing.
At the same time, the strain fields in device 3 are strong enough to modulate Tc by almost a factor of 4, from 200 to 780 mK. This finding suggests that the strain field spatially modulates the degree of 4f hybridization across the device and thereby also affects Tc. This mechanism, which is initially surprising, is compatible with experimental observations in the related compound CeRhIn5 in which hydrostatic pressure suppresses antiferromagnetism and eventually induces superconductivity. Despite the clear changes in the 4f magnetism and the spin fluctuation spectrum, the quantum oscillation frequencies remain unchanged in the entire pressure range up to the quantum critical point (24). However, the 4f hybridization increases, as evidenced by a quasiparticle effective mass that grows in response to applied pressure. This result strongly suggests that the hybridization with the 4f electrons varies without changes in the overall volume of the Fermi surface. Here we propose that the same microscopic physics underlies the spatial modulation of Tc in CeIrIn5 microstructures.
In general, strongly correlated materials exhibit a pronounced sensitivity to perturbations, owing to the small energy scales defining their physics. The strain accessible by our fabrication approach is sufficient to substantially alter the electronic properties of these materials without introducing chemical disorder. Unlike chemical approaches to tune correlations, the FIB provides micrometer-scale control over both the direction and magnitude of the induced strain field. We expect that the approach demonstrated here will enable spatial control of a wide range of broken symmetry states in strongly correlated systems. We envision clean interfaces between regions with different electronic order within a sample generated by a spatially modulated strain field—e.g., by generating superconducting regions in structures made from antiferromagnetic CeRhIn5 (24). Further, this approach is immediately compatible with any material that can be patterned using a FIB. Notably, although we focus on devices aligned along specific crystal axes, devices can be oriented along any direction to generate strain patterns of any desired symmetry. Strain engineering may offer an alternative way to fabricate superconducting circuitry within a metallic layer without any physical junctions, providing a route to fabricating superconductor/normal metal/superconductor Josephson junctions within a single crystal.
Supplementary Materials
science.sciencemag.org/content/366/6462/221/suppl/DC1
Materials and Methods
Supplementary Text
Figs. S1 to S13
Tables S1 and S2
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