Quantum effect-based flexible and transparent pressure sensors with ultrahigh sensitivity and sensing density

Although high-performance flexible pressure sensors have been extensively investigated in recent years owing to their diverse applications in biomedical and information technologies, fabricating ultrasensitive sensors with high pixel density based on current transduction mechanisms still remains great challenging. Herein, we demonstrate a design idea based on Fowler-Nordheim tunnelling effect for fabrication of pressure sensors with ultrahigh sensitivity and sensing density by spin-coating extremely low urchin-like hollow carbon spheres (less than 1.5 wt.%) dispersed in polydimethylsiloxane, which is distinct from the current transduction mechanisms. This sensor exhibits an ultrahigh sensitivity of 260.3 kPa−1 at 1 Pa, a proof-of-concept demonstration of a high sensing density of 400 cm−2, high transparency and temperature noninterference. In addition, it can be fabricated by an industrially viable and scalable spin-coating method, providing an efficient avenue for realizing large-scale production and application of ultrahigh sensitivity flexible pressure sensors on various surfaces and in in vivo environments.


Supplementary Tables
According to the effective medium theory (Equation 1), the current density J (x) increases with the mass fraction of conductive fillers x, which reflects the change in thickness when the composite is compressed due to external pressure. The black line in Supplementary Figure 1 shows the relationship between the readout signal and the pressure-induced deformation under the percolation effect.
where x is the mass fraction of conductivity fillers, J (x) is the current density of the composite, JM is the coefficient current density, xc is the percolation threshold, t = 1.6 for the threedimensional case.
Define ΔJ as the change of current density caused by external pressure, J0 as the initial current density, thus, Then, define a sample with a cross-section of S0 and height of d0, the volume V0=d0×S0. When the pressure applied, d would change Δd, then, Then the mass fraction can be described as follows, Thus, combine Equations 1-4, the relationship between J and Δd is as follows.
Then ΔJ can be described as follows, As strain ε is defined to be Δd/d0, the relationship between ΔJ and ε is as follows, It is usually a three-dimensional type in the application, so t=1. 6 and xc can be set as 28.95% when the filler is assumed to be perfect spheres 48 .

Supplementary Note 2. Relations between ΔRcon and ε of contact resistive.
The relationship between the contact resistance Rcon and the contact force F is described as equation 8 49 and accordingly, the relationship between the readout signal and pressure-induced deformation is shown by the red line in Supplementary Figure 1.
where Rcon is the contact resistance, k is a coefficient related to contact materials, F is the contact force, m is determined by the contact form (empirical studies have shown that when the contact form is point-type, m= 0.5; when it is line-type, m is between 0.5-1, approximately 0.7; when the contact form is face-type, m= 1).
While using E, S0, d0 to represent the modulus, cross-section area, and total thickness respectively. When the sample was pressed with a force F, its thickness will change Δd, then F can be described as, Thus, the ΔRcon and Δd can be as follows, As strain ε is defined to be Δd/d0, the relationship between ΔRcon and ε is as follows, To compare the slope with other two mechanisms, Equation 11 can be changed with mirror symmetry and translational symmetry, define ε' =1-ε, then the relationship between ΔRcon and ε' is as follows, It is usually a face type in the application, so m= 1.

Supplementary Note 3. Relations between ΔJ and ε of F-N tunnelling effect.
According to the F-N tunnelling equation described below, where A and B are empirical constants (A>0, B<0). J is a function of Ed, which is the electric field between two neighbouring UHCS in this study.
Ed is defined as (E is the electric potential, d is the thickness), Δd can be defined as follows, Combine Equations 13-15 to get the relationship between J and Δd, As strain ε is defined to be Δd/d0, the relationship between ΔJ and ε is as follows (A>0 and B<0),

Supplementary Note 4. Advantages of preloading for composite pressure sensors.
There are two reasons why a preloading process is needed in our manufacturing process: First, the sensing film and the electrodes are fabricated independently, so a preloading force is beneficial for ensuring good contact between the electrodes and the sensing film. Secondly, the sensor film is basically a polymer composite and it is widely recognized that the filler concentration in a polymer composite may vary from sample to sample even within the same batch during manufacturing. Given that the sensitivity of the composite film is highly dependent on the filler concentration, this variation in filler concentration may lead to an inconsistent sensitivity of the produced sensors. Therefore, in the fabrication process, we reduced the concentration slightly below the optimum concentration to avoid the inconsistency in sensor performance. Furthermore, by preloading process, the filler concentration can be tuned to make sure every sensor has the same sensitivity. Therefore, the preloading process is helpful in the sensor's performance and reliability. The preloading force can be controlled in the packaging process or tuned by the user before measurement. Of course, this may add an additional step in practical applications.

Supplementary Note 5. Basic requirements of the pressure sensing film for injection application in in vivo.
There are three prerequisites for the PDMS-based thin-film pressure sensors to be used in the implantable area.
First, the film should be flexible and thin enough to ensure a large sensing area after selfunfolding. A sensor array film should be large enough to cover the target organ in practical applications. In order to facilitate the implantation of the sensor array, it can be folded into a small part and be injected to the body. For example, to insert a 10 × 10 mm sensing film into a needle with diameter of 1.54 mm, the film should be folded at least for 5 times. To achieve the 5 times folding, the film should be thinner than 49.7 μm that can be calculated by Equation 18 50 .
where W is the width of a square piece of film with a thickness of t, and n is the desired number of folds to be carried out along alternate directions. Our sensing film is 20 μm thick and flexible, which allows sufficient folding before inserting into the syringe needle.
Secondly, the film should have the ability to unfolded with no damage after being folded and injected. Most flexible sensors with ultra-high sensitivity based on micro/nano structure usually cannot be bended for 180° for multiple times, which will damage these structures. Additionally, the injected sensor should be able to unfold in seconds. When immersed in water, our folded sensor quickly unfolded within 9 s ( Fig. 3a and Supplementary Movie 2), demonstrating its potential for injection into the body and in vivo self-unfolding to support large-area detection.
Thirdly, the implantable sensor should overcome the in vivo isostatic pressure. Supplementary Figure 15b shows two adjacent spheres at a critical position where spheres A and B can just contact with each other. By drawing a circle with point E as the centre and CE as the radius, a round area is formed. If the projection of all the sphere centres on the horizontal plane falls within this area, they will be in contact with their neighbours no matter how they moved horizontally. We define CE as the effective radius (rer), thus a "cylinder 1 (Supplementary Figure   15c)" can be built with a radius of (rcylinder1=rer+rUHCS+lspine), in which all the (n0+1) spheres will always be in contact with each other.
If spheres with a diameter of 600 nm are used, AB and AC are the radii of a hollow sphere (300 nm). BE is the spine length (80 nm). So, the effective radius can be calculated as follows: The concentration of spheres in the cylinder 1 of Supplementary Figure 15c   consideration. td is defined as the nearest distance between two adjacent spheres when they are not in direct contact. Thus, td /2 could be regarded as the extension of the radius with UHCS. In this situation, the cylinder diameter dcylinder2 is increased compared with cylinder 1, and considering that the td is in the range of 0-30 nm of 600 nm diameter spheres, here we may use the following expression to calculate the diameter of cylinder1 with a parameter td.
Then the solutions to Equation 37 are As there are three real roots, in this study, td is the distance between two spines, so we can choose the root 0 <xi< 600 as the root for td.
On the other hand, according to the electric-field characteristics on the point of a conical conductor 51 , the electric potential difference is, where ΔU is the electric potential difference, C is a constant determined by the electric and geometric parameters, r is the distance, λ is a constant determined by geometric parameters.
As shown in Supplementary Figure 16b, A  The parameter λ can be calculated as follows 51 Thus, when the centre of the second sphere drops in the yellow circle, the probability of the two UHCS contacting each other is 50%. All the UHCSs whose centre drops on the yellow circle will form the big black section, which defines the border of cylinder 3, as shown in Supplementary   Figure 17.