Japanese Team Develops First-Principles Method to Calculate Superconductor Coherence and Penetration Lengths Without Experimental Inputs

Predicting the temperature at which a material becomes superconducting from its crystal structure and constituent elements is a major challenge in condensed matter physics and computational materials science. Over the past 25 years, the accuracy of superconducting transition temperature ($T_c$) predictions has steadily improved, and the approach has been used to search for hydrides that superconduct under high pressure. But the transition temperature alone is not enough for researchers developing new superconducting materials, or for engineers designing superconducting magnets for MRI scanners and particle accelerators. To know how strong a magnetic field a material can withstand, and how much current it can carry before losing superconductivity, other physical quantities are needed.

The key ones are the coherence length (${\xi }_0$), which characterizes how the superconducting state varies in space, and the London penetration depth (${\lambda }_L$), which describes how deeply an external magnetic field penetrates from the surface of a superconductor. The ratio of these two lengths is an important indicator for distinguishing type-I superconductors, which expel magnetic fields, from type-II superconductors, which let quantized magnetic flux enter their interior and can maintain superconductivity even in relatively strong fields. Practical devices such as superconducting magnets mainly use type-II superconductors, yet conventional first-principles calculations lacked an established way to obtain both lengths from the same theoretical framework without relying on experimental values or empirical parameters.

The research group that took on this problem includes Associate Professor Mitsuaki Kawamura of the Faculty of Engineering at Yokohama National University, Associate Professor Takuya Nomoto of the Graduate School of Science at Tokyo Metropolitan University, Dr. Niklas Witt of the University of Würzburg, and Professor Ryotaro Arita of the Graduate School of Science at the University of Tokyo. The team extended superconducting density functional theory to develop a first-principles method that calculates the coherence length and penetration depth without experimental inputs. According to the team, it is the first method to obtain the superconducting transition temperature and both lengths from a single theoretical framework. However, the achievement is the development of a computational method and its validation against existing experimental data. It does not involve synthesizing a new superconducting material or demonstrating a practical device.

Introducing spatial modulation of Cooper pairs to calculate the superconductor's response

Conventional superconducting density functional theory (SCDFT) describes the superconducting state by assuming that Cooper pairs, in which electrons bind together, are distributed uniformly throughout the crystal. This framework is effective for predicting transition temperatures, but if only uniform states are treated, it is difficult to directly obtain the coherence length, which characterizes spatial variation of the superconducting order, or the penetration depth, which represents the response of the currents that screen an external magnetic field. Calculating these quantities requires examining how the superconducting state responds to spatial variation and to current.

To overcome this limitation, the team applied a gradual spatial modulation to the macroscopic wave function of the Cooper pairs. Specifically, they gave the center of mass of the Cooper pairs a finite momentum $Q$ and calculated states in which the superconducting order parameter varies in space. In this way they obtained, from first principles, the superconducting gap $\Delta (Q)$, which reflects the strength of electron pairing, and the supercurrent density $J(Q)$ as functions of temperature $T$ and momentum $Q$. Analyzing these two results made it possible to derive length scales that were previously hard to evaluate directly.

The coherence length ${\xi }_0$ is determined by examining how the superconducting gap decays under spatial modulation. For niobium (Nb), for example, the team identified the momentum at which the averaged superconducting gap falls to $1/\sqrt{2}$ of its value at $Q=0$, and evaluated ${\xi }_0$ by extrapolating the temperature dependence. This analysis uses relations based on Ginzburg–Landau (GL) theory. However, these relations hold strictly only near the superconducting transition temperature $T_c$, so evaluating the coherence length at absolute zero involves theoretical approximations.

The penetration depth ${\lambda }_L$, meanwhile, can be obtained in two ways. One is to calculate it directly from the initial slope of the supercurrent density $J(Q)$ as the momentum $Q$ increases. The other is to find the "depairing current," the limit at which current destroys the superconducting state, from the point where $J(Q)$ peaks, and combine it with the coherence length ${\xi }_0$ to work backward. Note that these two are not independent experimental verifications; they evaluate data from the same first-principles calculation with different theoretical methods.

Numerical difficulty was also a major challenge in this work. According to the team's release, the relevant length scales reach 100 to 1,000 times the lattice constant of the crystal. However, the publicly available materials we could check do not give specific lattice constant and length pairs corresponding to this ratio. Handling such long-range spatial variation in reciprocal (wave-number) space requires precisely calculating tiny momenta of less than 0.1% of the crystal's reciprocal lattice vectors. The team addressed this by building a calculation scheme that stably samples extremely small momenta and converges the electronic states self-consistently.

"Parameter-free" here means that no experimental values are used as adjustable inputs when predicting the transition temperature and the two lengths. The actual calculations still require theoretical approximations and numerical settings, so it does not mean that they can be performed without assuming any conditions. When checking the accuracy of the results, the team compared them with existing experimental data.

Validation on eight superconductors, with niobium agreeing well with experiment

To check the accuracy of the new method, the team performed calculations on eight known superconductors. They comprise six elemental metals — aluminum (Al), niobium (Nb), tin (Sn), indium (In), tantalum (Ta) and lead (Pb) — plus the A15 compound ${\mathrm{V}}_3\mathrm{Si}$ and hydrogen sulfide (${\mathrm{H}}_3\mathrm{S}$), which shows a high transition temperature under ultrahigh pressure of about 200 GPa. All are classified as conventional superconductors, in which Cooper pairs form through electron interactions mediated by lattice vibrations (phonons).

Results varied by material, but values close to experiment were obtained for several systems. For niobium, the calculated transition temperature was 8.09 K, relatively close to the experimental 9.50 K. The coherence length ${\xi }_0$ was calculated at 34 nm against an experimental 39±1 nm, and the penetration depth ${\lambda }_L$ at 34–40 nm against experimental values of 27–39 nm cited in the paper. The classification into type I and type II, based on the GL parameter $\kappa ={\lambda }_L/\xi$ (the ratio of penetration depth to coherence length), also matched the experimental classification for all eight materials examined. In GL theory, the boundary between type I and type II is given by $\kappa =1/\sqrt{2}$.

• Calculated
• Experimental (representative)

データを表で見る

	Calculated (Units differ by item)	Experimental (representative) (Units differ by item)
Transition temperature Tc (K)	8.09	9.5
Coherence length ξ0 (nm)	34	39
Penetration depth λL (nm)	37	33

However, not all materials achieved the same level of accuracy. For aluminum, which has an extremely low transition temperature, the calculated value was 1.61 K against an experimental 1.140 K, and the coherence length differed by more than a factor of two, at 636 nm calculated versus 1,600 nm experimental. The authors attribute the discrepancy in part to the difficulty of precisely calculating electronic states and electron–phonon interactions in the very small energy range below a millielectronvolt. Although errors of around a factor of two remain for some materials, the achievement is that the method captured the order of magnitude of the length scales and the trends between materials without tuning parameters to experimental values.

The calculated and experimental results for three key materials are summarized below.

Material / conditions	First-principles values	Experimental comparison values	Assessment and limitations
Niobium (Nb) / ambient pressure, body-centered cubic	$T_c=8.09,\mathrm{K}$, ${\xi }_0=34,\mathrm{nm}$, ${\lambda }_L=34\text{–}40,\mathrm{nm}$	$T_c=9.50,\mathrm{K}$, ${\xi }_0=39\pm 1,\mathrm{nm}$, ${\lambda }_L=27\text{–}39,\mathrm{nm}$	Transition temperature is lower than experiment, but the two lengths agree relatively well. Classification as a type-II superconductor also matches.
Aluminum (Al) / ambient pressure, face-centered cubic	$T_c=1.61,\mathrm{K}$, ${\xi }_0=636,\mathrm{nm}$, ${\lambda }_L=19\text{–}20,\mathrm{nm}$	$T_c=1.140,\mathrm{K}$, ${\xi }_0=1600,\mathrm{nm}$, ${\lambda }_L=16,\mathrm{nm}$	Captures its type-I character, but a discrepancy of more than a factor of two remains in the coherence length.
Hydrogen sulfide (${\mathrm{H}}_3\mathrm{S}$) / 200 GPa, $Im\bar{3}m$ structure	$T_c=181,\mathrm{K}$, ${\xi }_0=3.0,\mathrm{nm}$, ${\lambda }_L=19\text{–}22,\mathrm{nm}$	$T_c=184,\mathrm{K}$, ${\xi }_0\approx 2,\mathrm{nm}$, ${\lambda }_L\approx 20,\mathrm{nm}$ or $37,\mathrm{nm}$	Predicts values under ultrahigh pressure. Experimental penetration depths differ by measurement method, and anharmonic vibrations are not included.

It is worth noting that this validation is not a statistical evaluation of predictive performance over many randomly selected materials. The targets are eight materials chosen from known conventional superconductors, so the results do not guarantee universal predictive accuracy for unknown materials. The ranges of experimental values in the table mainly reflect measurement error and differences in measurement methods; they do not indicate the error range of the computational method itself.

Applied to hydrogen sulfide under ultrahigh pressure, predicting properties hard to measure

The new method is likely to be especially valuable for superconductors in extreme environments where experimental measurement is difficult. A prime example is hydrogen sulfide (${\mathrm{H}}_3\mathrm{S}$), for which superconductivity at a high transition temperature of about 200 K was reported in 2015. Obtaining the high-temperature superconducting state in ${\mathrm{H}}_3\mathrm{S}$ requires an ultrahigh pressure of about 200 GPa (roughly 2 million atmospheres). This is comparable to the pressure at the Earth's center and cannot be reproduced with ordinary experimental equipment.

Such experiments use a pressure device called a diamond anvil cell (DAC) to compress tiny samples of a few tens of micrometers or less. Because the sample is extremely small and surrounded by the pressure apparatus, it is not easy to control the magnetic field precisely or to measure the penetration depth directly. Assuming a crystal structure with $Im\bar{3}m$ symmetry, the team obtained a transition temperature of 181 K, a coherence length of 3.0 nm and a penetration depth of 19–22 nm. The transition temperature is close to the experimental 184 K, and the penetration depth is consistent with some experimental results.

This result does, however, have theoretical limitations. The ${\mathrm{H}}_3\mathrm{S}$ calculation does not account for anharmonicity of the lattice vibrations. In materials containing light elements such as hydrogen, quantum fluctuations of atomic nuclei and anharmonicity of lattice vibrations are known to strongly affect crystal stability and electron–phonon interactions. The authors also list incorporating anharmonic effects as a future theoretical extension.

The experimental values used for comparison are also uncertain. The penetration depth measurements cited in the paper split into about 20 nm, obtained from magnetization curves, and about 37 nm, estimated from the decay of trapped magnetic flux. The sample inside a diamond anvil cell has a complex shape, and the geometric corrections needed for magnetic measurements are difficult, so interpretation of the results carries a margin. It is significant that the calculated 19–22 nm agrees well with one experimental result, but pinning down the true penetration depth will require further experimental verification.

The depairing current differs from the critical current of practical wires

Another important point is how to interpret the "depairing current" obtainable with this method. The depairing current represents the theoretical limit at which current can no longer sustain Cooper pairs in an ideal, uniform superconductor, with no impurities, crystal defects or grain boundaries. Using the coherence length and penetration depth, the depairing current density can be expressed roughly as follows.

Here, ${\Phi }_0$ is the magnetic flux quantum and ${\mu }_0$ is the vacuum permeability. This relation shows that if the penetration depth and coherence length can be obtained from first-principles calculations, a guide to the current density an ideal superconductor can withstand can also be evaluated. However, the "critical current" that indicates the current-carrying performance of practical wires must not be equated with this theoretical depairing current.

When current flows through a type-II superconductor, the Lorentz force acts on the quantized flux lines that have entered its interior. Movement of flux lines produces electrical resistance and heat, so keeping the superconducting state stable requires "pinning," which fixes flux lines using crystal defects, precipitates and the like. The critical current of practical materials depends strongly on flux-line motion, grain boundaries and microstructure, and is often far lower than the ideal depairing current.

In fact, the calculated depairing current density for niobium shown in the paper was $19\times 10^7,A/c{\mathrm{m}}^2$, whereas the measured critical current density used for comparison was $0.75\times 10^7,A/c{\mathrm{m}}^2$, a difference of about 25 times. This gap shows that even if first-principles calculations can predict the limit of an ideal crystal, they cannot directly predict how much current an actual wire can carry. For applications to electromagnets and power transmission cables, the effects of crystal defects and microstructure on current transport must be evaluated separately.

Reproducing the Uemura plot, with an eye on materials discovery

Using the calculated superconducting properties, the team also attempted to construct a "Uemura plot," which shows the relationship between the superconducting transition temperature $T_c$ and the Fermi temperature $T_F$ on a logarithmic scale. The Uemura plot is an empirical classification chart proposed by Professor Yasutomo Uemura of Columbia University and colleagues, based on penetration depth measurements by muon spin rotation ($\mu \mathrm{SR}$) and other data. Unconventional high-temperature superconductors such as cuprates and iron-based materials tend to cluster in the region where $T_c/T_F$ is about 0.01–0.1, while simple metallic superconductors are widely distributed in regions with lower ratios. This difference is an important clue for comparing the electronic states and pairing character of superconductors.

In this study, the team built the Uemura plot using only physical quantities obtained from first-principles calculations, with no experimental values as inputs. As a result, they reproduced the placement of elemental metals such as aluminum and niobium in the region corresponding to conventional superconductors. For ${\mathrm{V}}_3\mathrm{Si}$, which has a relatively high transition temperature, and for ${\mathrm{H}}_3\mathrm{S}$ under ultrahigh pressure, both the energy scale involved in Cooper pair formation and the "phase stiffness," which expresses the resistance of the superconducting order to phase variation, tended to be large.

Phase stiffness is related to quantities such as the superconducting electron density and the Fermi temperature, and is an important concept for considering the stability of the superconducting state. The results suggest that treating the transition temperature and penetration depth within the same first-principles framework may allow the relationships among such quantities to be analyzed from a microscopic viewpoint. However, what was confirmed is a correlation in the calculations; a new mechanism that causes high-temperature superconductivity has not been demonstrated.

As future applications, the team cites the development of superconducting wires and electromagnets, the search for materials for fusion reactors, MRI scanners and quantum computers, and the screening of new hydrides aimed at room-temperature superconductivity. If not only the transition temperature but also the response to magnetic fields and the ideal limit of current density can be calculated, there will be more information on which to compare candidate materials. However, no such devices were designed or built in this study, and practical use still faces engineering challenges not covered by the calculations, including grain-boundary control, mechanical strength, thermal conductivity and chemical stability.

The results were published on November 3, 2026, in the peer-reviewed journal Physical Review Letters, volume 137, article 196001, and were selected as an Editors' Suggestion, a category that highlights notable papers. An explanatory article was also published in Physics, the American Physical Society's web magazine. These indicate that the novelty and scholarly importance of the work were recognized editorially, but they do not mean that independent replication by third parties has been completed.

The next challenge is to extend the method to a wider range of superconductors. At present it targets conventional superconductors based on electron–phonon interactions and cannot be applied as is to unconventional superconductors, such as cuprates, nickelates and heavy-fermion systems, where strong electron correlation is important. Theoretical extensions are also needed to reduce computational cost and to handle anharmonicity of lattice vibrations and complex anisotropy. The result is an important step toward predicting from first principles not just the superconducting transition temperature but also the response to magnetic fields and currents. If predictive accuracy for unknown materials and independent verification accumulate, the prospects for computational searches for superconducting materials will widen further.