Curvature programmed inkjet printing enables adaptive deposition for Gaussian sintering lasers

Abstract

The inherent Gaussian intensity distribution of laser beams causes critical issues such as ablation, spattering, and porosity during laser sintering and melting. Although modulating the intensity distribution by a beam shaper can alleviate these problems, it suffers from high cost, limited durability, and significant energy loss. To address these challenges, we propose a thickness-matching strategy that aligns film geometry with the laser intensity profile. Mathematical analysis provides the optimal thickness distribution, while inkjet printing with computable parameters enables curved circuits with ideal profiles. The curved profile strategy demonstrates enhanced performance in both semiconducting and metallic circuits. For indium tin oxide (ITO) conductive glass, up to a 3.8-fold improvement in conductance and a 5.1% increase in transmittance are achieved compared with planar circuits, while for copper (Cu) conformal circuits, the electrical conductivity is improved by 160% compared with planar circuits. This work establishes a practical additive manufacturing route for high-performance conformal circuits.

To address non-uniform sintering caused by Gaussian laser beams, the authors use adaptive inkjet printing to form curved circuits that match the light profile, thereby enhancing the electrical and optical performance of metal and semiconductor films.

Introduction

Selective laser sintering, as a rapid densification process of micro- and nanoparticles, has emerged as a critical technique for powder bed fusion and flexible electronics fabrication owing to its small radiation area, high power density, and controllable scanning paths^1–3. Aiming to approach the high mechanical or electrical performance of bulk materials, investigations have been conducted to minimize the defects and improve the uniformity of the laser sintered materials^4–9.

Sintering uniformity is crucial for enhancing component density and minimizing defects such as keyholes^10,11, ablation^12,13, and spattering^14,15. These issues primarily stem from the inherent non-uniformity in the spot energy distribution of single-mode lasers^16,17, which account for over 90% of industrial applications. Laser beam shaping is a straightforward solution, which includes techniques such as focusing, beam broadening, masking¹⁸, and converting Gaussian beams into locally uniform profiles (e.g., Bessel and rectangular beams¹⁹). However, laser beam shaping methods suffer from higher energy loss in the optical system (exceed 35%)^20,21, as shown in Fig. S1 in supplementary information, limited stability and lifespan of shaping elements²², higher system complexity and lower compatibility with laser galvanometer systems, which poses challenges for large-scale industrial applications²³.

In flexible electronics and powder bed fusion, the influence of laser parameters on sintering has been extensively studied. As laser energy input increases, nanoparticles undergo necking, coarsening, and ablation, which peak during the roughening stage and then decline due to material degradation^24,25. To precisely control the sintering degree, most studies have focused on optimizing laser parameters, such as beam diameter and power, to identify suitable conditions for different materials^26,27. More recently, to break through the performance limit, attention has shifted to the impact of film properties, particularly thickness, on grain coarsening, leading to preliminary models predicting sintering behavior as a function of thickness². However, existing research remains largely limited to planar films^2,15,28, while the interaction of Gaussian beams with inhomogeneous curved films presents unexplored opportunities for advancing laser sintering strategies (Fig. 1a).

Fig. 1. Curved circuitry better matched to the laser’s Gaussian intensity profile.

a In conventional flat deposition, Gaussian laser sintering produces a non-uniformly sintered circuit, with beam center oversintered and ablated while edges remain undersintered. b By tailoring the deposition profile to match the Gaussian beam, a uniformly sintered circuit can be obtained, achieving uniform sintering without changing the beam shape. c Inkjet-printing process from I, liquid column, to II, uniform unit circuit, and III, shape-matched curved circuit illustrating the stacking of uniform unit circuits to form a curved profile.

Alternatively, in this work, we propose a method to improve the sintering uniformity induced by the natural, widely-used non-uniform laser. Unlike the complex and costly laser beam shaping, we propose a strategy to adapt the thickness of the deposition layer to the laser beam energy profile, which is simple and applicable, as illustrated in Fig. 1b, c. Therefore, the objective of the current work is to present proof of concept and to demonstrate the potential of this approach in real applications such as conductive glasses and conformal circuits.

Results

Gaussian energy distribution of a normal scanning laser beam

It is well recognized that during propagation, a laser beam contracts to its smallest diameter and highest intensity at the beam waist before expanding again. This behavior arises from the Gaussian nature of the beam, where focusing or diffraction effects cause convergence near the focal point, followed by gradual divergence due to diffraction²⁹. Based on this feature, researchers modulate the beam diameter (D) by controlling the lens-to-object distance (z). Here, the laser beam used in this study had a focal length of L_f = 125 mm and for the convenience of sintering, the lens-to-object distance is always larger than the focal length, L_f. To quantify beam expansion beyond the focal point, we measured the beam diameter D as a function of axial distance z within the range of 125 ≤ z ≤ 155 mm (Fig. 2a). In this beam-expanding region (z > L_f), the beam diameter changes linearly with beam distance with a relation of D = 0.042 ⋅ z − 5.15 (in mm), as indicated by the red line in Fig. 2a. In this study, the beam diameter D is defined as the average of two orthogonal diameters (D_x and D_y), as shown in Fig. 2b. Within the range of 125 ≤ z ≤ 155 mm, the two diameters exhibit a high degree of overlap, with a spot symmetry index of $\mid M_x^2-M_y^2\mid \le 0.08$, which indicates that the laser beam used in this study maintains excellent axial symmetry³⁰.

Fig. 2. Laser beam characterization and laser intensity integration.

a The laser beam expansion regions can be accurately modeled using the linear function. b The laser beam exhibits excellent quality and axial symmetry. c Beam intensity distribution for different power levels at fixed diameter. d Beam intensity distribution for different diameter levels at fixed power. e The energy distribution is obtained by integrating the beam intensity over the scanning path.

To adapt the deposition thickness with the laser profile, one needs to obtain the laser profile of the laser used for sintering. Thus, a beam quality analyzer is employed to measure the exact energy profile of a typical single-mode laser. Figure 2c illustrates the measured energy profiles at different laser powers of 2, 4, 6, and 8 W, respectively. The beam distance (z) is fixed at 135 mm, which gives the beam diameter of around 520 ± 10 μm (1/e² beam diameter). Consistent with previous studies[30-31], the results show that the radial laser intensity profile agrees well with Gaussian curve (Eq. 1) with a determination coefficient (R²) of 0.998. The laser intensity distribution, I(x, y), as shown in the inset of Fig. 2c, can be described as follows³¹:

$$I\left(x,y\right)=I_0\exp \left(-2\frac{x^2+y^2}{w^2}\right)=I_0\exp \left(-8\frac{x^2+y^2}{D^2}\right)$$ 1

where I₀ is the peak intensity at the beam center, x and y are the transverse coordinates, w and D are defined as the 1/e² radius and diameters, respectively. Additionally, beam intensity profiles were measured at various diameters, and the coefficient of determination (R²) for the Gaussian fits of the energy profiles was at least 0.997 within the beam diameter range of 320 μm ≤ D ≤ 1500 μm (125 mm ≤ z ≤ 155 mm), confirming the beam’s stable Gaussian characteristics at different spot sizes, as shown in Fig. 2d. The corresponding fitting parameters are summarized in Table S1 of the Supplementary Information.

In the laser sintering process, the laser is normally scanning through the materials (Fig. 2e). Assuming that the scanning length is significantly larger than the width of the laser beam, one can calculate the energy profile of the accumulated energy in the following manner. In the case that the laser moves in the x-direction, the mean projected laser energy on the substrate in the y-direction can be calculated as follows:

$$E_{\mathrm{s}}\left(y\right)={\int }_{-\infty }^{\infty }I\left(x,y\right)dx=I_0\cdot \exp \left(-8\frac{y^2}{D^2}\right)\cdot {\left(\frac{\pi D^2}{8}\right)}^{0.5}$$ 2

As a result, the energy distribution after laser scanning depends only on the y-coordinate (Fig. 2e) and can be well described by a Gaussian function (a comparison between the laser intensity distribution and the resulting energy distribution is illustrated in Fig. S2).

Desired material deposition profiles adapted to Gaussian laser

To adapt the profile of deposition thickness, h(y), with laser energy distribution, E_s(y), one needs to find out the relation, E_s = f(h). A simplistic assumption is that the energy density is proportional to the mass to be sintered. However, our previous study shows that the relation between E_s and h is relatively complex due to the complex laser penetration, reflection, and absorption mechanisms². Therefore, based on a heat transfer analysis, the following equation was developed:

$$E_{\mathrm{s}}=f\left(h\right)=\frac{\mathrm{\Delta }H\rho h}{\eta \alpha }\cdot \frac{\exp \left(-\mu h\right)+1+\left(1-\alpha \right)\exp \left(-\mu h\right)}{1-\exp \left(-\mu h\right)}$$ 3

where h is deposition thickness, η is the ratio of the energy conducted to the deposited nanoparticles to the overall energy absorbed by the substrate, the value of η is dependent on the physical properties of the substrate and is approximately 0.6 for glass substrates (calculated with the theoretical formula for the heat conduction of a semi-infinite body with transient heat conduction under Dirichlet boundary conditions, detailed in supplementary information, pages S12–S13). And μ, ΔH, ρ are the attenuation coefficient, enthalpy, and density of nanoparticle deposition, α is the absorbance of the substrate. The derivation of Eq. (3) is provided in detail in our previous study².

Once the laser energy distribution is known, the ideal corresponding thickness of the deposition layer can be obtained by finding the inverse function of E_s = f(h)

$$h=f^{-1}\left(E_{\mathrm{s}}\right)$$ 4

However, due to the complexity of the function of E_s = f(h), it is difficult to have the explicit analytical expression of f⁻¹(E_s). Alternatively, one can get h by means of iterative calculation with the following iteration equation, h_n+1 = g(h_n), which is derived from Eq. (4).

$$h_{n+1}=g\left(h_n\right)=\frac{E_{\mathrm{s}}\cdot \eta \alpha \cdot \frac{\exp \left(-\mu h_n\right)+1+\left(1-\alpha \right)\exp \left(-\mu h_n\right)}{1-\exp \left(-\mu h_n\right)}{h}_n}{\mathrm{\Delta }H\rho }$$ 5

Therefore, by discretizing E_s(y) at selected values, which are uniformly sampled at discrete intervals within the range of  − D to  + D, and applying Eq. (2) and Eq. (5), the deposition thickness h(y) at each discrete y position is calculated using the fixed-point iteration based on Eq. (5). Finally, the thickness distribution is obtained by fitting the discrete results with the Gaussian function as follows:

$$h\left(y\right)=c_1\exp \left(-\frac{y^2}{c_2}\right)$$ 6

where c₁ and c₂ are coefficients to be determined. For the Gaussian curve, h, a peak value of A_calc, width of W_calc (1/e² width), it can be expressed as Eq. (7), and the maximum thickness A_calc and the width W_calc of the deposition layer can be calculated as

$$h\left(y\right)=A_{\mathrm{calc}}\exp \left(-\frac{8y^2}{W_{\mathrm{calc}}^2}\right)$$ 7

$$A_{\mathrm{calc}}=c_1;W_{\mathrm{calc}}=\sqrt{8c_2}$$ 8

Here, we have developed the mathematical process to calculate the desired profile of deposition layer that fits the laser energy distribution. For the laser we used in the lab, which has a laser power of P = 4 W and a beam diameter of D = 520 μm, the best match is a Gaussian deposition profile with a maximum thickness and a width of A_calc = 14.5 μm and W_calc = 235 μm (once the laser parameters are modified, the calculated profile changes, as shown in Fig. S3).

Fabrication of coffee-ring-suppressed unit circuits in inkjet printing

Although Eq. (7) provides the theoretically optimal profile, fabricating and precisely controlling the circuit profile remains a significant challenge. Given that the traditional film deposition techniques, such as physical and chemical vapor deposition, are normally employed for depositing films with uniform thickness, here, we developed a calculable strategy that employs inkjet printing to deposit curved circuit through layer-by-layer stacking, such that their profiles match the laser distribution.

For mathematical tractability and computational feasibility of stacked unit circuits, uniform deposition of the unit circuit is highly desirable. However, spatially non-uniform evaporation during inkjet printing often results in the coffee-ring effect, a widely reported and well-known feature leading to non-uniform cross-sections^32,33. To address this issue, we extended our previous strategy of regulating single-drop deposition profiles by employing a binary solvent and an external temperature field³⁴ for the fabrication of unit circuits. Specifically, a binary solvent system composed of ethylene glycol (EG) and isopropyl alcohol (IPA) in a 1:1 volume ratio was adopted, and ink droplets were evaporated on a heated copper block under a controlled external temperature field. As the substrate temperature increased and sequentially reached the boiling points of IPA and EG, the relative strengths of the flows inside the liquid column changed significantly^32,34.

Figure 3 a illustrates the competing flows during droplet evaporation. Capillary flow (La) and thermal Marangoni flow (Ma_T) drive nanoparticles outward toward the three-phase contact line (TPCL) and thereby enhance the coffee-ring effect, while solutal Marangoni flow (Ma_C) transports nanoparticles inward, suppressing the effect. Based on flow theory, the strengths of these flows can be expressed as^35,36:

$$La=\frac{\rho l{\sigma }_{lv}}{{{\mu }_0}^2},M{a}_T=\frac{ld\sigma /dT}{2{\mu }_0\alpha }\mathrm{\Delta }T,M{a}_C=\frac{ld\sigma /dC}{2{\mu }_0{D}_f}\mathrm{\Delta }C$$ 9

where l is the circuit width, ΔC is the concentration difference between the droplet apex and the contact line, dσ/dC and dσ/dT are the variations of surface tension with concentration and temperature, μ₀ is the viscosity, and D_f is the diffusion coefficient of IPA in EG. In addition, ρ is the droplet density, σ_lv is the liquid–vapor surface tension, ΔT is the temperature difference between the droplet apex and the contact line, and α is the thermal diffusivity of the liquid.

Fig. 3. Mechanism and validation of uniform-deposition control via a binary solvent and external temperature field.

a Schematic of the competing flows in an evaporating droplet on a heated substrate: capillary flow (La), solutal Marangoni flow (Ma_C), and thermal Marangoni flow (Ma_T). b Theoretical prediction of the uniform-deposition window based on flow-strength analysis; uniformity is expected when the balance index 0.5 ≤ Ma_C/(Ma_T + La) ≤ 2. c ITO unit circuits printed on silicon at different substrate temperatures, confirming the predicted transition from ring-dominated to uniform deposition (scale bar: 100 μm). d Uniform unit circuits on different substrates (glass, silicon, superalloy) showing quasi-rectangular cross-sections (deviation η ≤ 3.7%).

According to theoretical analysis, uniform deposition is expected when Ma_C ≈ La + Ma_T (commonly taken as 0.5 ≤ Ma_C/(La + Ma_T) ≤ 2^34,37). To examine this condition, we calculated the values of Ma_C/(La + Ma_T) over the range 60 °C ≤ T ≤ 220 °C (see Table S2 in supplementary information for details). The results show that within 150 °C ≤ T ≤190 °C, the balance index indeed falls in the range 0.5 ≤ Ma_C/(La + Ma_T) ≤ 2, indicating effective suppression of the coffee-ring effect (Fig. 3b). To verify this theoretical prediction, we printed ITO unit circuits on single-crystal silicon substrates at different temperatures (see Fig. 3c). The results revealed that when T = 160 °C and T = 180 °C, the circuits exhibited nearly uniform deposition with minimal coffee-ring effect, in agreement with the theoretical expectation. Furthermore, ITO unit circuits were also printed on substrates with different wettabilities (glass, silicon, and superalloy) at T = 160 °C, and their cross-sections were measured, as shown in Fig. 3d. The results showed that regardless of the substrate material, the printed ITO circuit cross-sections could be regarded as quasi-rectangular with a deviation of η ≤ 3.7%, η is calculated by the following equation:

$$\eta =\frac{{\int }_0^{200}\mid z_1\left(x\right)-z_2\left(x\right)\mid dx}{S_{\mathrm{rec}}}$$ 10

where, z₁(x) and z₂(x) denote the measured and ideal rectangular profile heights at position x, respectively, and S_rec is the area of the reference rectangle.

Stacking-based inkjet-printing strategy for curved circuits with Gaussian-matched profiles

As mentioned above, by employing the binary solvent and external temperature field strategy, inkjet printing arranges a series of droplets into a straight line, resulting in a quasi-rectangular unit circuit with a width of w₀ and a thickness of h₀. By stacking unit circuits with a number of n and a displacement of d, which can be easily adjusted by the programs in the printer, we can fabricate deposition layers with different profiles in a shape-matched stacking manner, as illustrated in Fig. 4a. The profile of the stacked circuit can be manipulated by adjusting the printing parameters w₀, h₀, d, and n (Fig. 4b, c). The displacement d (Fig. 4c) and layer number n can be readily modified by the printer program. Furthermore, the width w₀ of the unit circuit decreases linearly with increasing substrate temperature, while the thickness h₀ increases inversely with increasing substrate temperature (Fig. 4d). Owing to mass conservation, it is expected that:

$$w_0{h}_0=C$$ 11

Fig. 4. Fabrication of curved circuits by stacking uniform unit circuits using inkjet printing.

a–c Curved profiles obtained by varying the unit width w₀ and stacking displacement d. d Printing temperature T controls w₀ and h₀ of the ITO unit circuit. e Cross-sectional area of the unit circuit (w₀h₀) at different substrate temperatures, showing that w₀h₀ is nearly constant. Data are shown as mean  ± s.d. (standard deviation error bars), with n = 3 independently fabricated unit circuits per temperature, and individual data points shown as dots. f Correspondence between theoretically calculated profile parameters and stacked circuit parameters. g Cross-section of the optimal deposition printed under the recommended parameters (d = 20 μm, w₀ = 140 μm) and the error E between the calculated and deposited profiles. h Among the nine curved circuits, the circuit fabricated with the theoretically recommended parameters shows the smallest E.

As shown in Fig. 4e, this relation is in good agreement with experimental data (w₀ and h₀ were measured from surface optical images and cross-sectional SEM, as shown in Fig. S4).

For the stacked circuit, the width (W_print), thickness (A_print), as shown in Fig. 4f, can be estimated as follows (when n < w₀/d):

$$W_{\mathrm{print}}=w_0+\left(n-1\right)d$$ 12

$$A_{\mathrm{print}}=n{h}_0$$ 13

For pairing a target Gaussian profile of deposition layer with a printed stacked circuit (Fig. 4f) we matched the cross-sectional thickness, A_print, and the width, W_print, of the printed circuit with the A_calc and W_calc of the target Gaussian circuit (A_calc = A_print, W_calc = W_print). Here, the parameter W_calc in the Gaussian curve is the featured width, normally called 1/e² width^38,39, which represents the concentration of the curve after filtering out the secondary regions.

By associating the Eqs. (11)–(13), one can calculate the desired inkjet printing parameters for matching the laser with a profile described with A_calc and W_calc (see Eqs. (6)–(8)) as follows,

$$w_0=\frac{nC}{A_{\mathrm{calc}}}$$ 14

$$d=\frac{A_{\mathrm{calc}}{W}_{\mathrm{calc}}-nC}{A_{\mathrm{calc}}\left(n-1\right)}$$ 15

To calculate the printing parameters w₀ and d, one needs to know the values of A_calc, W_calc, C and n. Among these, A_calc and W_calc are the coefficients of the model-predicted Gaussian profile of the deposition layer, which is the best-match of a given laser. According to Eqs. (5)–(8), as described in Section 2.2, it can be determined that A_calc = 14.5 μm and W_calc = 235 μm for a laser with P = 4 W and D = 520 μm. Besides, the constant C ≈ 340 μm² was determined from the scanning electron microscope (SEM) cross-sectional image of the ITO unit circuit (Fig. S4). The parameter n, representing the number of stacked layers, can be estimated from the maximum target height of the deposition profile, A_calc, and the thickness of each printed layer, h₀. Measurements indicate that the thickness of a single uniform ITO unit circuit is 2.1–2.8 μm, depending on the substrate temperature (Fig. S4). The target stacked height is A_print = A_calc = 14.5 μm, thereby allowing the number of layers to be approximated as n = int[A_calc/h₀], yielding n = 6 or 7, with n = 6 adopted in this study. Now, we have determined all the required parameters as follows, A_calc = 14.5 μm, W_calc = 235 μm, C = 340 μm² and n = 6. According to Eqs. (14)–(15), the inkjet printing parameters can be calculated as d = 18.9 μm and w₀ = 140.7 μm. For the convenience of printing, they are rounded to d ≈ 20 μm and w₀ ≈ 140 μm.

To validate the theoretical model, the stacked circuits were printed using the recommended parameters (d = 20 μm, w₀ = 140 μm) and the cross-sectional morphology, labeled as the optimal profile (opt.), were examined using scanning electron microscopy (SEM), as shown in the inset of Fig. 4g. It can be seen that the profile of the printed circuit agrees reasonably with the desired function, indicating that the printing model works well.

To quantitatively evaluate the difference of the profiles between the target and the printed one, a parameter E is defined as follows:

$$E=\frac{{\int }_{-\infty }^{\infty }\mid h_{\mathrm{print}}\left(y\right)-h_{\mathrm{calc}}\left(y\right)\mid dy}{S_{\mathrm{calc}}}=\frac{{\int }_{-\infty }^{\infty }\mid h_{\mathrm{print}}\left(y\right)-h_{\mathrm{calc}}\left(y\right)\mid dy}{\sqrt{\frac{\pi }{8}}{A}_{\mathrm{calc}}{W}_{\mathrm{calc}}}$$ 16

where h_print(y) and h_calc(y) are the variations of printed and model-calculated heights, respectively. Having noted that the printed curve may be partially higher or lower than the calculated one, as shown in the inset image of Fig. 4g, the absolute difference between two heights is used in Eq. (16). According to Fig. 4h, a shifting of the printing parameters, such as droplet spacing and unit circuit width, from the model-predicted ones would result in a deviation of the printed circuits away from the target profile, which further proves the validity and accuracy of the proposed printing strategies (complete cross-sectional SEM images are available in Fig. S5).

To further proof the entire concept, the traditional uniform deposition and curved deposition were sintered with a laser with a Gaussian energy profile (P = 4 W, D = 520 μm, matching Eq. (7)). As shown in Fig. 5a and Fig. S6, for the planar deposition, the excessive energy concentration at the beam center causes severe ablation and the formation of coarse grains, while the insufficient energy at the periphery results in partially sintered or even unsintered nanograins; only a narrow intermediate zone exhibits moderate grain coarsening, which provides limited improvement in conductivity and transmittance. In contrast, Fig. 5b demonstrates that the curved deposition profile effectively redistributes the laser energy, enabling uniform sintering across the center, mid-span, and edge regions of the circuit; as a result, the grains are well connected throughout the film, leading to markedly enhanced electrical performance. Moreover, the X-ray diffraction (XRD) patterns in Fig. 5c further support these findings: compared with the planar circuit, the curved circuit exhibits sharper and narrower diffraction peaks, indicating a narrower grain size distribution and higher crystallinity⁴⁰.

Fig. 5. Comparison of sintering homogeneity between planar and profile-optimized curved ITO circuits.

a Gaussian laser sintering of planar deposition resulted in non-uniform circuits. b Gaussian laser sintering of profile-opt. curved deposition resulted in uniform circuits. c XRD patterns of laser-sintered planar and curved ITO circuits. Compared with the planar circuit, the curved circuit exhibits sharper and narrower diffraction peaks, indicating more homogeneous grain growth and higher crystallinity.

Application demonstration: from high-performance ITO conductive glass to Cu conformal circuits

Having noticed that the well-sintered ITO film could exhibit excellent conductivity and transmittance, we employed ITO conformal circuits on glass substrates as an example for demonstrating the application of the proposed strategy. It is well recognized that conductive glass conductance and transmittance are two key performance metrics^41,42. However, the simultaneous optimization of these properties is challenging due to their differing dependencies on processing parameters. Specifically, higher laser energy improves the overall sintering degree and electrical conductivity but induces localized ablation at the beam center, adversely affecting optical transmittance. Similarly, increasing film thickness lowers resistivity but may compromise visible-light transparency. In this context, the curved circuit strategy shows significant advantages in controlling film thickness and achieving uniform sintering without ablation, thereby enabling potential synergistic enhancement of both electrical and optical performance.

For electrical performance evaluation, a series of curved circuits (profile-opt. and other curve profiles) as well as uniform planar circuits (same mass of nanoparticles as the curved circuits kept) were printed and sintered. We used the profile deviation, E, from the model-predicted shape to index each circuit. The corresponding conductivity as well as the transmittance of the circuits were measured. As shown in Fig. 6a, the conductivity decreases monotonically with the parameter E, among which the optimum circuit achieves the highest conductivity (3.8-fold higher than that of planar circuits). In addition, the laser-sintered curved circuits exhibited superior stability in the accelerated thermal cycling test, with resistance increasing by less than 10% after 50 cycles, compared to 35% for the planar circuits (see Fig. S7).

Fig. 6. Performance advantages of the curved circuit strategy.

a, b Conductivity and transmittance of laser-sintered planar and curved (profile-opt. and other profiles) ITO circuits show that the profile deviation E is inversely correlated with both properties. c Transmittance-resistance mapping highlights the simultaneous enhancement of conductivity and transmittance achieved by the curved circuit strategy, in contrast to literature-reported trade-offs^45–47. d Profile-opt. curved ITO circuits exhibit higher directional transmittance than planar circuits over  − 90^∘ ≤ θ ≤ 90^∘. e Theoretically predicted optimal curved profiles for Ag, Cu, ITO, and SrTiO₃ under the same Gaussian beam, showing material-dependent differences in peak height and width. f Experimental validation for Cu confirms that the predicted optimal profile corresponds to the best sintering performance. g Comparison of Gaussian-beam curved circuits and beam-shaped planar circuits demonstrates the advantages of the curved circuit strategy: lower energy requirement, wider processing window, and higher achievable conductivity. h Benchmarking against thermal sintering on Al₂O₃ and polyimide substrates shows that laser processing avoids oxidation on high-melting substrates and prevents thermal damage on low-melting substrates, highlighting the broad adaptability of the curved circuit strategy. All fitted functions can be found on pages S11–S12 of the Supplementary Information.

For optical performance evaluation, as shown in Fig. 6b, we arranged curved ITO circuits on a glass substrate in a crisscross pattern with a uniform circuit spacing of 300 μm in both horizontal and vertical directions. The transmittance (angle of incidence, θ = 0^∘) of the conductive glass was subsequently measured. It shows that the monochromatic transmittance is almost identical for the radiation with a wavelength above 390 nm (inset of Fig. 6b). Moreover, the transmittance also decreases with an increase in E.

For the purpose of co-optimization of electrical and optical performance, the data were plotted in a space of transmittance-resistance, as shown in Fig. 6c. Here, the reciprocal of sheet resistance (1/R_s has units of □/Ω) is employed as the horizontal variable^43,44, and the transmittance of λ = 550 nm radiation at an incident angle of θ = 0^∘ is employed as the vertical variable. The data in literature were also added in Fig. 6c for comparisons^45–47, and it could be seen that the elevation of transmittance was normally achieved by trading-off the electrical conductivity. However, with the strategy proposed in this work, high transmittance and high conductivity can be simultaneously achieved. Say, the curved circuit exhibits the best synergistic performance of ITO conductive glass (1/R_s > 0.05 □/Ω, transmittance  > 92.5% at λ = 550 nm) compared to the cases reported in literature to the best of our knowledge. Interestingly, due to the moth-eye-like morphology of the curved circuits, the directional transmittance can be significantly improved, especially for the incidence angle higher than 45^∘, as shown in Fig. 6d.

In addition, we further extended the theoretical model to different materials. Figure 6e shows the optimal predicted curved profiles for Ag, Cu, ITO, and SrTiO₃ under the same Gaussian beam, confirming that materials with higher reflectivity and thermal conductivity (e.g., Ag, Cu) require narrower and taller optimal profiles, while ITO and SrTiO₃ exhibit relatively broader and flatter shapes. For Cu circuits, experimental validation demonstrates that the predicted optimal profile indeed corresponds to the best sintering performance, as reflected by the highest conductivity at minimal profile deviation E (Fig. 6f). Furthermore, we also fabricated a large-scale conformal circuit sample (Fig. S8), which exhibited an average conductivity as high as 1.3 × 10⁵ S ⋅ cm⁻¹, representing a 1.6-fold improvement compared with planar circuits and further demonstrating the scalability and robustness of the proposed strategy.

Furthermore, by systematically comparing Gaussian-beam sintering of curved circuits with beam-shaped planar circuits, we identified several clear advantages of the proposed strategy (Fig. 6g). The required energy to achieve optimal sintering is lower, the processing window is wider as indicated by the larger area where conductivity exceeds 10⁴ S ⋅ cm⁻¹, and the maximum achievable performance is superior, avoiding the local cracking issues often observed in sharp-edged planar circuits.

Finally, Fig. 6h benchmarks laser sintering against traditional thermal sintering on different substrates. For high-melting substrates such as Al₂O₃, the short sintering time of laser processing minimizes oxidation and leads to better electrical performance. For low-melting substrates such as polyimide, the localized heating nature of the laser prevents thermal damage to the substrate (as shown in Fig. 6h, PI films can only withstand sintering up to 200 ^∘C), further underscoring the broad applicability and superiority of the laser approach.

In summary, the curved circuit strategy shows high tolerance to both circuit and substrate materials, simultaneously improving the transmittance and sheet conductance of ITO conductive glass, as well as the conductivity of Cu circuits.

Discussion

Direct sintering using Gaussian laser beams often causes central ablation and insufficient edge sintering, while existing beam-shaping techniques are complex, energy-intensive, and costly. To address these issues, we propose a method of using curved circuits to match the Gaussian laser distribution for fabricating homogeneous circuits. The key findings are summarized below:

1. A theoretical model is developed for calculating the best-match deposition profile. Calculated deposition profiles exhibit a Gaussian distribution, but differ from the laser intensity distribution.

2. A printing strategy is proposed for fabricating the desired deposition with a Gaussian profile. The desired curved circuits were fabricated through stacking of uniform unit circuits, enabled by balancing the three concurrent flows within the ink. The printed circuit profiles have an average deviation E < 1.7% from the theoretical Gaussian profile and exhibit consistent grain coarsening after sintering.

3. Demonstrative cases of functional circuits are presented. For ITO conductive glass, the curved circuit strategy enables co-optimization of electrical-optical performance. For metallic Cu, curved circuits achieve higher conductivity and broader processing tolerance compared with both planar circuits fabricated by beam-shaping optics and conventional thermal sintering, highlighting its broad applicability.

Limitations and outlook. Despite these advances, the curved-circuit strategy still faces several challenges for industrial deployment. In terms of profile-design freedom of circuits, it will be necessary to introduce digitally programmable control of laser spot size and intensity so that the beam profile can be adapted to target geometries of curved circuits⁴⁸, thereby overcoming the constraints of the laser on the profile of curved circuits. In terms of packaging and 3D stacking, compatibility can be realized through post-sintering planarization, in which the temporary curvature formed during sintering is eliminated and the circuits are transformed into uniform-thickness functional layers with tolerances controllable to below 1 μm by employing advanced low-temperature calendaring or ultra-thin coating methods⁴⁹. In terms of processing speed, our current demonstrations are limited to laboratory-scale printing speeds (approximately 10 mm/s), and achieving m/s-level throughputs may require addressing droplet stability at high actuation rates, maintaining suppression of coffee-ring flows under shortened drying times, and incorporating in situ thermal/optical feedback for closed-loop control of the sintering window. In terms of high-frequency signal integrity, while this study focused on direct current and low-frequency performance, future work will extend to GHz-frequency characterization in order to validate that the improved conductivity uniformity can be translated into stable RF signal integrity, which is essential for industrial applications.

Methods

Nano-ink preparation

Conformal circuits were fabricated on glass, Al₂O₃ ceramic substrates, and polyimide (PI) films using ITO and Cu nanoparticles via inkjet-based additive manufacturing. First, nanoparticles were dispersed in a mixed solvent to formulate the nanoinks for inkjet printing. The ITO nanoparticles (purity = 99.9 wt%, Deke Daojin, China; average particle size = 30 nm) were dispersed in a binary solvent composed of ethylene glycol (purity =  99.9 wt%, Macklin, China) and isopropanol (purity = 99.9 wt%, Macklin, China) at a volume ratio of 1:1, with a nanoparticle concentration of 200 mg/mL. Polyvinyl pyrrolidone (PVP) powder (purity = 99.99 wt%, molecular weight = 58,000 g/mol, Macklin, China) was added at 2.5 mg/mL as a steric stabilizer for the suspension. The mixture was mechanically dispersed using an ultrasonic cell crusher (Scientz-IID, Scientz, China) operated at 150 W for 180 min, while the dispersion temperature was maintained at 10 °C using a water bath. After sonication, the ITO nanoink was passed through a polytetrafluoroethylene (PTFE) filter (pore size = 0.45 μm) to remove large aggregates. The physical properties of the ITO ink were adjusted to typical ranges suitable for stable inkjet printing, with a surface tension of 35–45 mN/m.

The Cu nanoink was prepared following the same formulation, dispersion, and filtration protocol as the ITO nanoink. Specifically, Cu nanoparticles (purity = 99.9 wt%, Deke Daojin, China; average particle size = 30 nm) were dispersed in the identical ethylene glycol/isopropanol (1:1 v/v) solvent mixture at a concentration of 200  mg/mL, with PVP added at 2.5 mg/mL as stabilizer. The suspension was ultrasonically dispersed at 150 W for 180 min at 10 °C and subsequently filtered through a 0.45 μm PTFE membrane prior to printing.

Inkjet printing

The ITO and Cu nanoinks were deposited as droplets (nominal diameter = 20 μm) onto flat SiO₂ glass, Al₂O₃ ceramic, PI film, single-crystal silicon, and superalloy substrates using an inkjet printer with a positioning repeatability of  ± 5 μm (Jetlab 4xl, MicroFab, USA). Prior to printing, all substrates were cleaned in an ultrasonic bath using isopropyl alcohol, ethanol, and ultrapure water sequentially, each for 3 min, and then dried in a hot-air oven. To promote solvent evaporation and improve deposition uniformity, the substrate temperature was controlled in the range of 140–180 °C during printing using a proportional-integral-derivative (PID) controller. The ambient relative humidity was maintained between 30% and 80%, and the surface energy of the cleaned substrates was controlled in the range of 25–45 mN/m to ensure stable droplet spreading and coalescence.

Laser sintering

The deposited nanoparticle films were sintered using a continuous-wave ytterbium-doped fiber laser (MFSC75, Max Photonics, China). The laser beam was steered by a programmable galvanometer scanner system (RC1001, Jinhaichuang, China), which allowed precise control of the laser power and scanning speed along predefined trajectories. The beam diameter and intensity distribution at the sample plane were characterized using a beam profiler (SP620U, Ophir Optronics, Israel) and used to determine the effective spot size and energy density for the laser sintering process.

Characterizations and testing

The characteristics of the fabricated curved circuits were evaluated using a series of optical, structural, and electrical measurements. An optical super-field microscope (DSX1000, Olympus, Japan) was employed to observe the macroscopic morphology of the films, while field-emission scanning electron microscopy (FE-SEM, SU8020, Hitachi, Japan) was used to examine the microstructure. The circuit resistance was measured using a direct current resistance tester (HPS2661, Helpass, China). The optical transmittance of conductive glass samples was measured using a UV-vis-NIR spectrophotometer (UV-3600, Shimadzu, Japan).

X-ray diffraction (XRD) patterns were recorded on a diffractometer (D8 Advance, Bruker, Germany) equipped with a Cu Kα source (wavelength = 1.5406 Å), operated at 40 kV and 40 mA. The scans were carried out in step-scan mode with a step size of 0.02° and a scan rate of 4°/min. Thermal cycling tests were conducted by heating the glass substrate with a thin-film heater (LP-24, Dongtai, China), while the temperature was monitored with a K-type thermocouple. As illustrated in Fig. S7, each cycle consisted of rapid heating from room temperature to 150 °C, a short isothermal hold at 150 °C, and subsequent cooling to near room temperature by natural air convection.

The laser beam was shaped using a multi-element optical path, as schematically shown in Fig. S1. The raw Gaussian beam first passed through an iris-diaphragm aperture stop (Iris Diaphragm, Thorlabs, USA), which introduced an energy loss exceeding 15%, and was then focused onto a circularly serrated aperture apodizer (Serrated Aperture, LightTrans, Germany) to further tailor the intensity distribution, with an additional loss of more than 20%. Finally, a collimating lens assembly (Collimating Optics, Thorlabs, USA) was used to form a flat-top beam profile on the substrate.

For comparison, thermal sintering experiments were conducted in a tube furnace (BTF-1400C, BEQ, China) to evaluate the electrical performance of thermally sintered circuits. The samples were heated at a rate of 10 °C/min to the target temperature, held isothermally for 30 min, and then cooled naturally in air to below 75 °C before electrical measurements.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Author contributions

X.C., Z.T., and L.Q. conceived and supervised the project. X.C. proposed the concept of curvature programmed circuits, performed the inkjet printing and laser sintering experiments, prepared the figures, and wrote the draft of the manuscript. X.C., M.Z., and L.Q. jointly developed and formalized the curvature programmed inkjet printing strategy for curved circuits. M.Z. refined the coffee-ring suppression strategy and its theoretical description. J.Z., Z.T., and L.Q. edited the manuscript and provided helpful discussions. J.Z., Z.T., L.Q., and X.C. acquired and managed the funding for this project. All authors commented on and approved the submission of this work.

Peer review

Peer review information

Nature Communications thanks Bowen Zhu, Jin Huang, Angelos Markopoulos, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

Source Data are provided with this paper. The data generated in this study are provided in the Supplementary Information and Source Data files. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-68613-y.