Self-Limited Formation of Bowl-Shaped Nanopores for Directional DNA Translocation

Solid-state nanopores of on-demand dimensions and shape can facilitate desired sensor functions. However, reproducible fabrication of arrayed nanopores of predefined dimensions remains challenging despite numerous techniques explored. Here, bowl-shaped nanopores combining properties of ultrathin membrane and tapering geometry are manufactured using a self-limiting process developed on the basis of standard silicon technology. The upper opening of the bowl-nanopores is 60–120 nm in diameter, and the bottom orifice reaches sub-5 nm. Current–voltage characteristics of the fabricated bowl-nanopores display insignificant rectification indicating weak ionic selectivity, in accordance to numerical simulations showing minor differences in electric field and ionic velocity upon the reversal of bias voltages. Simulations reveal, concomitantly, high-momentum electroosmotic flow downward along the concave nanopore sidewall. Collisions between the left and right tributaries over the bottom orifice drive the electroosmotic flow both up into the nanopore and down out of the nanopore through the orifice. The resultant asymmetry in electrophoretic–electroosmotic force is considered the cause responsible for the experimentally observed strong directionality in λ-DNA translocation with larger amplitude, longer duration, and higher frequencies for the downward movements from the upper opening than the upward ones from the orifice. Thus, the resourceful silicon nanofabrication technology is shown to enable nanopore designs toward enriching sensor applications.


Table of Contents:
Supporting Note S1. Resistance model for BNP pore size and surface charge extraction.
Supporting Note S2. Dominance of electrophoretic force during dsDNA translocation.
Supporting Note S3 Estimation of capture radius of nanopore for DNA translocation              S3 Supporting Note S1. Resistance model for extraction of the BNP size and surface charge Following the concept of our previous publication 1 , the keystone to establish the resistance model for the BNPs is to calculate the effective transport length, Leff. The Leff of a nanopore is defined as the distance between the two points along the central axis of the nanopore where the electric field intensity is e -1 of its maximum value.
As there is neither source nor sink along the path of electric field lines apart from the surface charge, the integration of electric flied flux on any equi-electric-field-intensity-surface should be the same. Thus, the task to find the position of an equi-electric-field-intensity-surface where the electric field is e -1 of its maximum can be converted to finding an equi-electricfield-intensity-surface whose area is e times the area of the smallest restriction of the nanopore. Figure S4. Calculation of Leff of the BNPs.
As shown in Fig. S4, the equi-electric-field-intensity-surface in a bowl can be approximated as the hemisphere interception with the hemisphere of the bowl. Assume that the centre of the equi-electric-field-intensity hemisphere is located at the bottom centre of the bowl. From the geometrical relationship, the area of the intercepted equi-electric-field-intensity hemisphere can be expressed as as the area of the smallest restriction of the BNP where the electric field reaches its maximum. dp is the diameter of this smallest constriction in circular shape.
x can be found by substituting Eqs. S1 and S2 into Eq. S3. One of the real roots of this thirdorder equation is the desired value. By considering the electric field below the smallest restriction, the total Leff becomes: Hence, the resistance of the BNP is where, ρ is the resistivity of the electrolyte.

Extraction of surface charge density from conductance measurement
According to the literature, the conductance of a nanopore can be contributed by two parts: bulk conductance Gb and surface conductance Gs as 2-4 : wherein, Gb=1/R while Gs is determined by the surface charge density and corresponding ion mobility 3,4 : where, σ is the surface charge density and μ the mobility of ions in the diffuse layer of EDL. However, the BNP has an uneven cross-sectional area. Leff is used instead of h in Eq. S7 for estimation: By fitting the conductance G vs. conductivity 1/ρ data with Eq. S8, dp and σ can be extracted.

S5
Supporting Figure

Dominant factor
Electrophoretic force: where, q is the charge of the translocating object and Ez the electric field intensity along the vertical direction, i.e. z-direction.
Electroosmotic force: where, A is the surface area of the object, η the viscosity of water, w the vertical component of the water velocity and n the normal of the translocating object surface. Consider one base pair of a double-stranded DNA (ds-DNA), its side area A that interacts with EOF is: where, r=1.1 nm is the radius of DNA and ds=0.33 nm the length of one DNA base pair. The velocity gradient of EOF can be approximated by wmax/λD where wmax is the maximum vertical component of the water velocity and λD the Debye length. In 500 mM KCl, λD=0.4 nm.
The effective charge of ds-DNA is reduced from -2e/bp to -0.49e/bp by considering the screening effect from the electrolyte. Thus, the total force acting on one base pair of ds-DNA is: For a 3 nm-diameter BNP, the distribution of total force on such a base pair is shown in Fig.  S6. The movement of DNA is governed by the electrophoretic force due to its high charge density. For larger BNPs, the EOF effect is weaker while the electrophoretic force stays unchanged. Thus, the DNA is always driven by the electrophoretic force. Figure S6. Distribution of the total force F=Fel+Feof acting on a base pair of ds-DNA at (left) V=+500 mV and (right) V=-500 mV in a BNP of dp=3 nm in 500 mM KCl.

Competition between electrophoretic force and electroosmotic force
In the case with a BNP of dp=3 nm, the maximum value of the electric field is 7×10 7 V/m. Thus, the maximum electrophoretic force is Fel=0.49×1.6×10 -19 ×7×10 7 =5.5 pN.
The maximum water flow rate is 0.35 m/s, and the maximum electroosmotic force is Feof =1.9 pN according to Eq. S10, which is 35% of Fel. It is worth noting that Feof of the BNP is almost 10 times larger than that in a counterpart TCP also of dp=3 nm, because the EOF rate is of the order of 0.01 m/s.
For a better comparison, the Fel and Feof on a base pair of ds-DNA is plotted along the central axis of the BNP of dp=3 nm, as shown in Fig. S7. Although the electrophoretic force governs the DNA translocation, the electroosmotic force is of the same order of magnitude and is anticipated to play a significant role in modulating the movement of DNA.  For the TCP in Fig. S8a, the electroosmotic vortexes are found neither in the lower reservoir near the smallest restriction of the nanopore at positive bias nor inside the nanopore above the smallest restriction at negative bias. Clearly different from BNPs, the EOF along the surface of the sidewall can spread to and fill up the smallest restriction of the TCP for both bias conditions and thereby contribute to the electroosmotic force that opposes the DNA translocation. However, a stream of water flow retarding the DNA translocation appears along the central axis of the BNP above and below the smallest restriction at negative and positive bias, respectively. For a better comparison between the driving forces in BNP and TCP, the maximum ratio of electroosmotic force to electrophoretic force along the central axes of the nanopores is shown in Fig. S8b at different bias voltages. The difference in this ratio at positive and negative biases is much larger for the TCP than for the BNP. For the TCP at positive bias seen in Fig. S8a, the EOF generated along the sidewall is seamlessly connected to the lower reservoir without any counter flow. This smooth connection leads to a higher velocity of EOF and correspondingly a stronger electroosmotic force than those in the counterpart BNP. Furthermore, higher biases result in larger maximum ratios for the BNP, which is significantly different from the TCP.

S9
Supporting Figure

Supporting Note S3. Estimation of capture radius of nanopore for DNA translocation
The effective capture radius (r*) of the analyte translocation can be defined as 7 where, dp is the pore diameter, h the thickness of the pore and ΔV the voltage applied to the electrodes, µ the DNA electrophoretic mobility, and D the DNA diffusion coefficient. D/µ is the potential where the capture hemisphere locates, i.e., V(r*) = D/µ. Considering a segment of dsDNA with its persistence length, 35 nm, 6  In this way, V(r*) = D/µ = 5.4×10 -4 V. For the λ-DNA translocation through an 8 nm BNP that only the thickness near the orifice is taken into consideration, the radius of capture hemisphere is around 1.2 µm, i.e., 120 nm at 100 mV and 0.6 µm at 500 mV. In the simulation, we can directly measure the radius of capture hemisphere, which is just the radius of equipotential surface of D/µ = 5.4×10 -4 V. As shown in Fig. S10, the capture radius r* is around 0.7 µm at -500 mV and 0.6 µm at +500 mV, which coincides with the theoretical estimation very well. It also indicates higher capture rate (translocation frequency) at negative bias.  Translocation experiments were first carried out using the dp=4.4 nm BNP (Fig. S11a) with λ-DNA. Typical translocation spikes in Fig. S11a could be observed only at negative biases, which can be accounted for by invoking the 4-fold difference in the ratio of total force at negative bias to that at positive bias (Fig. 4c in the main text). Frequently observed translocation waveforms displayed in Fig. S11b are characterized by a long low-level blockage followed by a short high-level one falling directly back to the baseline. The 16-μmlong λ-DNA with 48,502 base pairs soaked in electrolytes can convolute into a loose clump of 1.2 μm in gyration diameter. 5 With a persistence length of 35 nm typical for doublestranded DNA, 6 part of the λ-DNA can dwell in the bowl and occupies a considerable volume in the high-electric-field region of the BNP. This behavior can render a significant semiblockade responsible for the long low-level blockage in Fig. S11b. Although the 35-nm persistence length makes bending of the λ-DNA unlikely to fit into the 4.4 nm pore, the λ-DNA may slightly move and rotate until eventually threading through the BNP and causing the short high-level blockage. Lacking such a dynamic mechanism in the absence of a similar bowl below the BNP can explain no observed translocation from the lower reservoir, at positive bias.