Supporting Information for: Empirical Mappings of the Frequency Response of an Electron Ratchet to the

Characteristics of the Polymer Transport Layer

Mohamad S. Kodaimati1,2, Ofer Kedem1,2, George C. Schatz1,2* and Emily A. Weiss1,2* 1 Center for Bio-Inspired Energy Science, Northwestern University, 303 E. Superior Street, 11th floor, Chicago, Illinois 60611-3015 2 Department of Chemistry, Northwestern University, 2145 Sheridan Rd., Evanston, IL 60208-3113 *co-corresponding authors. Emails: e-weiss@northwestern.edu, g-schatz@northwestern.edu

Device Fabrication. We fabricated ratchet devices with similar structures to our previously

reported work on electron ratchets.1 Briefly, we describe the fabrication process below. We used

2” p-type (B-doped) Si wafers with 1000 nm of wet thermal oxide (NOVA Electronic Materials)

as a substrate. We spin-coat Shipley Microposit S1813 photoresist on the wafer for 45 s at 6000

rpm with a 2000 rpm/s ramp and subsequently bake the wafer for 1 min at 110°C. Using a maskless

aligner (Heidelberg µPG501 Maskless Aligner) we defined the contact lines (10 µm) and contact

pads (500x500 µm2 and 200x200 µm2). We then developed the wafer in MF319 developer for ~30

s. We then deposit 10 nm Ti followed by 40 nm Au using an AJA electron-beam physical vapor

depositor. We performed lift-off by sonication in acetone.

To deposit the Pt finger electrode (FE) array, we used focused-ion beam (FIB) deposition of

Pt using an FEI HELIOS Nanolab 600 FIB operated using Ga+ ions at 30 kV and 48 pA of current.

The electrodes were deposited with asymmetric thickness profiles and were 400 nm wide, 57.25

µm long with 400 nm spacing between electrodes. The electrodes were deposited as twelve 50 µm

sections which included an 8 µm long Pt cross-bar connecting sections of finger electrodes. We

characterize the FEs using a Bruker Dimension FastScan atomic force microscope (AFM), Figure

S1.

To deposit the dielectric layer encapsulating the FEs, we deposit a layer of plasma enhanced

chemical vapor deposition (PECVD) SiO2 followed by a layer of spin-on-glass (SOG) and a final

layer of PECVD SiO2. In both PECVD steps, we deposited ~20 nm of SiO2 using an STS LpX

PECVD (900 mTorr, 1420 sccm N2O, 500 sccm SiH4, 30 W HF, 20 sec, 300 °C plate, 250 °C

showerhead temp, and a chamber wall temperature of 75°C). We deposited the SOG by spin-

coated a 50%v of 11F SOG glass (Filmtronics) in isopropanol at 3000 rpm for 5 s (2000 rpm/s

ramp). We let the wafer rest in the spin-coated for 9 s and then subsequently bake the wafer for 1

min at 85 °C, 155 °C, and 255 °C. We then repeated the spin-coating and baking steps to deposit

a second layer of SOG. The wafer was then cured for 1 hr at 425°C in an N2 atmosphere using an

AS-Micro Rapid Thermal Processor. The resulting SOG layer was ~140 nm thick. The dielectric

layer thicknesses were characterized using a Filmetric F20 Thin Film Analyzer reflectometer. To

re-expose the FE contact pad, we perform another photolithography step to expose the FE pad and

etched the dielectric layer above the pad using a Samco RIE-10NR reactive ion etcher (20 sccm

CF4, 3.3 PA, 50W, 9 min). The photoresist was dissolved in acetone.

Using a photolithography procedure similar to what was described above, we deposited two

50-nm thick (10 nm Ti/40 nm Ag), 500 µm long contacts as source and drain electrodes straddling

the FE array. The wafers were sonicated in acetone and dried with nitrogen before being placed in

a 220°C oven for 1 hr. We silanized the wafer by submersing the wafer in 0.1 %v solution of

trimethoxy(octyl) silane (Sigma-Aldrich) in dry heptane overnight in an N2 box.

We deposited P3HT on our devices by spincoating a 7 mg/mL solution of P3HT (95-98%

regioregular, 20 kDA, American Dye Source) in 94% CHCl3, 5% dichlorobenzene, 1%

diiodoctane at 1000 rpm for 15 s in a saturated CHCl3 atmosphere. We left the film in the

spincoater for ~2 min and ~1 min in air before swabbing off the contact pads using CHCl3. We

then bake the wafer for 2 min at 90 °C. CS2 annealing was carried out by placing the P3HT film in

a recrystallization dish saturated with CS2 vapors. To remove the P3HT—in order to deposit a new

P3HT film on the same device—we sonicate the wafer in CHCl3.

Figure S1. Atomic force microscopy (AFM) linescans of the FEs of four devices D1 (A), D2 (B), D3 (C), and D4 (D).

Electrical Characterization. Electrical

connections to our devices were made use a

Signatone S-1160 probe station. Source-drain

current was measured using a Keysight

B2902A source-measure unit. The drain

electrode is connected to the common (chassis)

ground; the current is measured between the

source electrode and the common ground. We

applied a static DC bias to the FEs for the field-effect transistor (FET) mobility measurements to

the FEs using a Keithley 6430 Electrometer equipped with a pre-amplifier. During the ratchet

measurements, we applied an oscillating potential to the FEs using a Tektronix AFG3022C

function generator. All devices were tested prior to deposition of P3HT to insure that the source,

drain, and FE contacts all exhibited purely capacitive behavior.

From operating a ratchet device in a linear FET mode, we observe that the mobility of electrons

is a factor of 40 less than that of the holes. The linear FET mobility, 𝜇𝜇𝑙𝑙𝑙𝑙𝑙𝑙 , was calculated using eq

S1, where L and W are the length and width of the accumulation channel, Ci is the gate-channel

𝜇𝜇𝑙𝑙𝑙𝑙𝑙𝑙 =𝐿𝐿

𝑊𝑊𝐶𝐶𝑙𝑙𝑉𝑉𝑆𝑆𝑆𝑆�𝛿𝛿𝐼𝐼𝑆𝑆𝑆𝑆𝛿𝛿𝑉𝑉𝑔𝑔

� (𝑆𝑆1)

capacitance per unit area, VSD is the source-drain voltage, and �𝛿𝛿𝐼𝐼𝑆𝑆𝑆𝑆𝛿𝛿𝑉𝑉𝑔𝑔

�, is the derivative of the

source-drain current with respect to the gate voltage. �𝛿𝛿𝐼𝐼𝑆𝑆𝑆𝑆𝛿𝛿𝑉𝑉𝑔𝑔

� is calculated by fitting a line to the

gate voltage sweeps (as in Figure S6). The length and width of the channel are taken from optical

microscopy images and the capacitance per unit area was taken from our previously reported

values for similar devices1.

Figure S2. Sample FET mobility measurement of P3HT on a device where we sweep the gate voltage at VSD = -2 V

Tuning the Electrical Properties of P3HT. As discussed in the main text, we utilized two

methods to tune the electronic properties of the P3HT transport layer: i) illumination of the P3HT

film with 532 nm light and ii) annealing in the CS2 vapor. Devices were illuminated using a

Thorlabs 532 nm laser diode where the intensity was attenuated using a neutral-density wheel.

As expected, the conductivity of the P3HT films increased with increasing intensity of

illumination with a 532-nm laser diode. Figure S3A shows a representative set of ISD-VSD curves

for a single device at varying illumination intensities. We waited 10 min after beginning to

illuminate each sample before electrical measurements to insure that the sample reached a steady-

Figure S3. Characterization of the ISD-VSD curves (A), FET mobilities (black squares) and carrier concentrations (blue triangles) (B) of a single P3HT film on ratchet device D1 under increasing power of 532-nm laser illumination (0-120 mW/cm2

) without application of a ratchet potential.

state in terms of σ, µh, and nh. Figure S3B shows that, as we increase the illumination intensity,

the carrier concentration increases, but the FET mobility decreases. At the highest incident power

density (120 mW/cm2), the conductivity and carrier concentration of the sample are factors of

seven and 29 larger than in the dark, while the mobility is a factor of four lower than in the dark.

After the device was kept in the dark for several hours, the samples returned to their “dark states”

in terms of conductivity, carrier concentration, and mobility; this reversibility indicates that the

sample was not damaged during the photoexcitation. We observed similar trends in the mobility

and carrier concentration as a function of illumination intensity across P3HT layers spin-coated

upon the same device and across different devices. We attribute the increase in the carrier

concentration by photoexcitation to the photo-generation of excitons that undergo interchain

charge transfer to generate weakly bound polaron pairs which further dissociate into free charge

carriers within the P3HT film2-4. We suspect the decrease in the FET mobility upon illumination

is a result of exciton-exciton2, 5 and exciton-hole recombination2, 6, 7 resulting in a decrease in the

number of free carriers.

Since illumination modifies both the carrier concentration and mobility, we introduced CS2

annealing to deconvolve the effects of the carrier concentration and mobility on the observed

frequency response of the ratchet current. We placed the devices coated with P3HT films in a

crystallization dish saturated with CS2 vapor from 0 to 6 hours to disorder the morphology of our

P3HT films, see the SI for details. After six hours of CS2 annealing, the conductivity of the P3HT

film decreased by a factor of 60—caused by a decrease in the FET mobility by a factor of 74 while

the carrier concentration increased by a factor of 1.2, Figure S4.

In order to determine the origin of the decrease in mobility upon CS2 annealing, we used optical

spectroscopy to characterize P3HT films deposited on silanized SiO2/Si wafers—as a proxy for

our ratchet devices. Absorption measurements were carried out using a Varian Cary 5000

spectrometer in transmission mode with a 1-cm quartz cuvette for the solution samples and an

integrating sphere for the solid samples. We base-line corrected all spectra before measurement.

Photoluminescence spectra were collected using a Fluorolog-3 spectrofluorometer using either a

1-cm cuvette with a right-angle geometry (solution phase) or front-face geometry (solid phase).

Resonance Raman measurements were carried

out using a Horiba LabRam HR Evolution

confocal Raman microscope with 473 nm

excitation. The Raman measurements were

averaged over at least 10 different spots on the

P3HT film.

The absorbance of unaggregated P3HT in

CHCl3 has a broad peak centered at ~440 nm

consistent with literature results, Fig S5A.8

Upon spin-coating P3HT on a wafer, we

observe the formation of a dramatic

bathochromic shift in the absorbance along

with the formation of sharp distinct peaks—

consistent with the formation of J-aggregates

in highly crystalline P3HT.8 After CS2

annealing, we observe a concomitant decrease

Figure S4. Characterization of a ratchet device, D1, in the dark with increasing CS2 solvent annealing time without application of a FE potential A) ISD-VSD curves under varying annealing times. B) Plot of linear FET mobility (black, squares) and carrier concentration (blue, triangles) versus the CS2 annealing time.

in the J-aggregate absorbance and an increase in the monomeric absorption—indicating we are

disordering the initially crystalline P3HT film. We observed that as we increased the CS2

annealing, time we observe increasing monomeric features in the absorption spectra.

Photoluminescence (PL) measurements highlight a similar trend where we observe

bathochromically shifted PL for the spin-coated P3HT film (as compared to the unaggregated

P3HT in solution), Fig S5B. With increasing CS2 annealing, we observed an increased

hypsochromic shift in the PL spectra towards the monomeric absorption. Resonance Raman

measurements on a series of P3HT films support this hypothesis when we probe the symmetric in

plane ring C=C stretching mode at ~1445 cm-1. We observe a slight shift in the C=C mode (up to

3 cm-1) towards higher energies indicative of an increase in disorder in the P3HF films8, Fig S5C.

Through these three measurements, we consistently observe a decrease in the crystallinity with

CS2 annealing which accounts for the dramatic decrease in hole mobilities within these films.

Within the simulations and in our discussion of the experimental results, we assume the

morphology of the P3HT is uniform with a single effective mobility describing transport. While

highly ordered P3HT has anisotropic mobility ( a factor of 7 higher mobility for transport parallel

to the P3HT chains vs perpendicular9), the formation of highly ordered P3HT requires the use of

templating10, 11, electric fields12, or high purity (99%) regioregular P3HT9. Given our low

mobilities (~1 x 10-3 cm2 V-1s-1) and inability to resolve P3HT fibrils in AFM, we believe our films

are too disordered to have anisotropic mobilities.

Figure S5. Characterization of P3HT films deposited on SiO2/Si wafers with increasing CS2 solvent annealing time (0-12 hr) from black to red A) Absorption spectra of P3HT films (solid) and P3HT in CHCl3 (dashed). B) PL spectra of P3HT films (solid) and P3HT in CHCl3 (dashed) with 470 nm excitation. C) Resonance-Raman spectra of P3HT films with 473 nm excitation.

Effect of FE Shape on Ratcheting Frequency. To validate the effect of the FE shape on the

ratcheting frequency, we compared three devices with three different FE shapes at the same

mobility and carrier concentrations (±25%), Figure S6. We observed different frequency

dependences consistent with the carriers experiencing different potentials—and thus different drift

velocities—during the ratcheting mechanism. When comparing two different FE shapes (S1 and

S3), we observed that the linear relationship between carrier concentration and 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 is conserved

for devices with different potential shapes, Figure S7A. Furthermore, we observe the same 2/3

power relationship between the mobility and 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 , Figure S7B.

Figure S6. A) Plots of source-drain charge as a function of ratcheting frequency for devices with shapes S1 (blue), S2 (black), and S3 (yellow) 10 V sine waves with a fixed carrier concentration of 1.3×10-3 C cm-3 and mobility of 1.1×10-3 cm2 V-1s-1 . B) AFM linescan of the FEs of device S1. C) AFM linescan of the FEs of device S2. D) AFM linescan of the FEs of device S3.

Figure S7. A) Plot of peak ratcheting frequency vs. carrier concentration for devices with shape S3 with mobilities of 2.3x10-4 (black) and 1.3 x10-3 (red) cm2

V-1s-1. B) Plot of peak ratcheting frequency vs linear FET mobility for devices D1-4 with carrier concentrations of 1.7×10-3 (black) and of 3.7×10-3 (red) C cm-3.

Figure S8. A) Plot of peak ratcheting frequency divided by carrier concentration vs. mobility for devices D1-D4 with varying carrier concentrations. B) Plot of peak ratcheting frequency divided by 𝜇𝜇ℎ2/3 vs. mobility for devices D1-D4 with varying mobilities.

Effect of Carrier Concentration on Magnitude of Ratcheting Efficiency. As discussed in

the caption of Figure 3C (reproduced below as Figure S9C), we observed a monotonic decrease

in the magnitude of ratchet current per oscillation, 𝑞𝑞𝑜𝑜𝑜𝑜𝑜𝑜, with increasing carrier concentration, the

opposite of what we would expect for non-interacting particles, since more carriers available to

participate in the ratcheting mechanism should lead to higher ratchet current. Our observed trend

is therefore probably attributable to carrier-carrier interactions, specifically electron-hole

recombination2, that are encouraged by the ratcheting process. For example, during the oscillation

of the electric field, the holes (and electrons, to a lesser degree) are transported between the top

and bottom boundaries of the P3HT film. As the carriers transverse the center of the transport

layer, positively and negatively charged carriers are co-localized, transiently increasing the

probability of recombination events. These recombination events are second-order (at minimum)

and have a probability that scales as 𝑛𝑛ℎ2, so they result in a decrease in the ratchet current with

increasing carrier concentration. The trend of decreasing ratchet efficiency with increasing carrier

concentration is therefore likely an artifact of how we modulate carrier concentration: with light,

which creates both holes and electrons.

Bnv .

Figure S9. A) Plots of source-drain charge as a function of ratcheting frequency for device D1 driven by 10 V sine waves with a fixed carrier concentration of 2.6×10-3 ± 6.5×10-4 C cm-3, with differing mobilities. B) Plot of peak ratcheting frequency vs linear FET mobility for devices D1-4 with varying carrier concentrations. Solid traces are power law fits described in the main text. C) Plots of source-drain charge as a function of ratcheting frequency for device D1 driven by 10 V sine waves, with a fixed linear FET mobility of 1.0×10-4 cm2

V-1s-1 with differing carrier concentrations. D) Plot of peak ratcheting frequency vs. carrier concentration for devices D1-D4 with mobilities of 4.5×10-5 (yellow), 1×10-4 (black), 2.5×10-4 (red), and 5.8×10-4 (blue) cm2

V-1s-1. Data for a mobility of 5.8×10-4 cm2 V-1s-1

is only plotted if the frequency response had a single discernable peak, as opposed to two peaks.

1-Dimensional Overdamped Model. We adapted a model of a two-state potential by

Rozenbaum13 that we have shown to reasonably describe our experimental ratchet devices as a

reference for comparing against our experimental results. This deterministic model relies upon a

high-temperature expansion of the Smoluchowski equation for a time-dependent, spatially

periodic potential energy. The potential energy of the particles,

𝑉𝑉(𝑟𝑟, 𝑡𝑡) = 𝑓𝑓(𝑡𝑡) ∙ 𝑉𝑉(𝑥𝑥) (S2)

𝑉𝑉(𝑥𝑥) = 𝑝𝑝1𝑞𝑞

sin �2𝜋𝜋𝜋𝜋𝐿𝐿� + 𝑝𝑝2

𝑞𝑞sin �4𝜋𝜋𝜋𝜋

𝐿𝐿� (S3)

𝑓𝑓(𝑡𝑡) = �𝐴𝐴− for 𝑛𝑛𝑛𝑛 ≤ 𝑡𝑡 < 𝑛𝑛𝑛𝑛 + 𝑛𝑛−𝐴𝐴+ for 𝑛𝑛𝑛𝑛 + 𝑛𝑛− ≤ 𝑡𝑡 < (𝑛𝑛 + 1)𝑛𝑛 𝑛𝑛 ∈ ℤ (S4)

𝑤𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝐴𝐴+ = 1, 𝐴𝐴− = 𝛼𝛼 (−1 ≤ 𝛼𝛼 ≤ 1), 𝛿𝛿 ≡ 𝑛𝑛−/𝑛𝑛

𝑉𝑉(𝑟𝑟, 𝑡𝑡), is the product of a biharmonic spatial potential, 𝑉𝑉(𝑥𝑥), and temporal oscillations, 𝑓𝑓(𝑡𝑡),

between amplitudes 𝐴𝐴− and 𝐴𝐴+ with duty ratio, 𝛿𝛿. Here, we use a duty ratio of 0.5, 𝛼𝛼 of 0.5, with

𝑎𝑎1 and 𝑎𝑎2 of 0.25 V q and 0.008 V q. L is the periodicity of the potential, 800 nm. The particle

velocity is given by:

𝑣𝑣 = 𝜋𝜋4𝑆𝑆𝐿𝐿𝛽𝛽3𝑎𝑎12𝑎𝑎2(1 − 𝛼𝛼)2[(1 + 𝛼𝛼)Φ1(𝜉𝜉, 𝛿𝛿) + (1 − 𝛼𝛼)Φ2(𝜉𝜉, 𝛿𝛿)] (S5)

where D is the diffusion coefficient, 𝛽𝛽 = (𝑘𝑘B𝑇𝑇)−1 where 𝑘𝑘B is the Boltzmann constant and T is

the temperature (293 K), Φ1,2 are functions defined below, and 𝜉𝜉 ≡ � 𝐿𝐿2𝜋𝜋�2

/𝐷𝐷𝑛𝑛 is the dimensionless

oscillation frequency. We approximate the diffusion coefficient for our system using the Einstein-

Smoluchowski equation 𝜇𝜇h = 𝑞𝑞𝐷𝐷h/𝑘𝑘B𝑇𝑇, where 𝜇𝜇h is the hole mobility, 𝑞𝑞 is the elementary charge,

𝑘𝑘B is the Boltzmann constant and 𝑇𝑇 is the temperature.14 The current, 𝐼𝐼, is related to the particle

velocity, 𝑣𝑣, by

𝐼𝐼 = 𝑣𝑣𝑛𝑛e𝜎𝜎𝑞𝑞 (S6)

where 𝑛𝑛e is the carrier density, 𝜎𝜎 is the cross-sectional area of the source/drain electrodes, 25 um2,

and 𝑞𝑞 is the elementary charge, 1.6 ∙ 10−19 C. The Φ1,2 functions for deterministic switching are:

Φ1(𝜉𝜉, 𝛿𝛿) = 12𝜋𝜋2𝑥𝑥𝐶𝐶(𝑥𝑥,𝑦𝑦) (S7)

𝐶𝐶(𝑥𝑥,𝑦𝑦) = ∑ 𝐴𝐴(𝑥𝑥,𝑦𝑦,𝑛𝑛)∞𝑙𝑙=1 (S8)

𝐴𝐴(𝑥𝑥,𝑦𝑦,𝑛𝑛) = 𝜋𝜋�7+2𝜋𝜋2𝑙𝑙2�(1−cos𝑦𝑦𝑙𝑙)(1+𝜋𝜋2𝑙𝑙2)(16+𝜋𝜋2𝑙𝑙2) (S9)

Φ2(𝜉𝜉, 𝛿𝛿) = 12𝜋𝜋2𝑥𝑥𝑥𝑥(𝑥𝑥,𝑦𝑦) (S10)

𝑥𝑥(𝑥𝑥,𝑦𝑦) = (1 − 2𝛿𝛿)𝐶𝐶(𝑥𝑥,𝑦𝑦) − 12𝜋𝜋𝐷𝐷(𝑥𝑥, 𝑦𝑦) (S11)

𝐷𝐷(𝑥𝑥,𝑦𝑦) = ∑ ∑ [𝐵𝐵(𝑥𝑥,𝑦𝑦,𝑛𝑛,𝑚𝑚) + 𝐵𝐵(𝑥𝑥, 𝑦𝑦,𝑚𝑚,𝑛𝑛)]−1𝑚𝑚=−𝑙𝑙+1

∞𝑙𝑙=2 + ∑ 𝐵𝐵(𝑥𝑥,𝑦𝑦,𝑛𝑛,𝑚𝑚)∞

𝑙𝑙,𝑚𝑚=1 (S12)

𝐵𝐵(𝑥𝑥,𝑦𝑦,𝑛𝑛,𝑚𝑚) = 𝜋𝜋�5𝑙𝑙+(𝑙𝑙+𝑚𝑚)�4−𝜋𝜋2𝑙𝑙2��[sin𝑦𝑦𝑙𝑙(1−cos𝑦𝑦𝑚𝑚)+sin𝑦𝑦𝑚𝑚(1−cos𝑦𝑦𝑙𝑙)](𝑙𝑙+𝑚𝑚)𝑚𝑚(1+𝜋𝜋2𝑙𝑙2)(16+𝜋𝜋2𝑙𝑙2)[1+𝜋𝜋2(𝑙𝑙+𝑚𝑚)2] (S13)

where 𝑥𝑥 = 2𝜋𝜋𝜉𝜉, 𝑦𝑦 = 2𝜋𝜋𝛿𝛿.

We convert the calculated current to the charge by dividing by the oscillation frequency. We

observe that as we increase the mobility the peak oscillation frequency linearly increases, Figure

S10. Since the Smoluchowski equation, eq. s14, relates the flux, 𝐽𝐽(𝑥𝑥, 𝑡𝑡), to the derivative of the

particle potential energy,𝑈𝑈(𝑥𝑥, 𝑡𝑡), (with respect to the spatial coordinate) and particle

distribution, 𝜌𝜌(𝑥𝑥, 𝑡𝑡), incorporation of a spatially homogenous y-coordinate does not affect the

relationship between the diffusivity and peak frequency. 𝜕𝜕𝜕𝜕𝑡𝑡𝜌𝜌(𝑥𝑥, 𝑡𝑡) = −

𝜕𝜕𝜕𝜕𝑥𝑥

𝐽𝐽(𝑥𝑥, 𝑡𝑡) (𝑆𝑆14)

𝑤𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝐽𝐽(𝑥𝑥, 𝑡𝑡) = −𝐷𝐷 �𝛽𝛽𝜌𝜌(𝑥𝑥, 𝑡𝑡)𝜕𝜕𝜕𝜕𝑥𝑥

𝑈𝑈(𝑥𝑥, 𝑡𝑡) +𝜕𝜕𝜕𝜕𝑥𝑥

𝜌𝜌(𝑥𝑥, 𝑡𝑡)�

Finite-element Simulations of Charged-Particle Ratchets. We performed finite-element

simulations, using COMSOL Multiphysics 5.4, of an ensemble of non-interacting particles in a 4

µm (x) x 100 nm (z) box with periodic boundary conditions in the x-dimension and diffuse

scattering off the z boundaries. Particle transport is dictated by a Newton-Langevin equation, eq

s15, where m is the mass of the particle (1.26 x 10-18),

𝑚𝑚�̈�𝒓(𝑡𝑡) = −d𝑉𝑉(𝑡𝑡,𝒓𝒓)d𝒓𝒓

− 𝛾𝛾�̇�𝒓 + 𝐹𝐹B + 𝜉𝜉(𝑡𝑡) (S15)

Figure S10. A) Simulated plots of source-drain charge as a function of ratcheting frequency for devices increasing mobility of 1.1×10-3 cm2 V-1s-1 . B) Plot of peak ratcheting frequency vs mobility for experimentally measured devices D1-4 with carrier concentrations of 1.2×10-3 (black) and a 1D overdamped model (red). Details of the 1D model are found in the SI.

𝑉𝑉(𝑡𝑡, 𝒓𝒓) = 𝑥𝑥(𝑡𝑡)𝑈𝑈(𝒓𝒓) is the temporally, 𝑥𝑥(𝑡𝑡), and spatially, 𝑈𝑈(𝒓𝒓), varying potential; 𝛾𝛾 is the viscous

drag coefficient; 𝐹𝐹B describes collisions with boundaries; and 𝜉𝜉(𝑡𝑡) is a white noise term. The

viscous drag coefficient is 𝛾𝛾 = 6π𝜂𝜂𝑟𝑟p, where 𝜇𝜇 is the dynamic viscosity of the medium, and 𝑟𝑟p is

the radius of the particles (2.5 nm). The velocity immediately following a boundary scattering

event is defined by eq S16 , where 𝜃𝜃 = acosΓ − π2, Γ is a random number between -1 and 1, and

�̂�𝑡 and 𝑛𝑛� are unit vectors, tangential and normal to the boundary, respectively.

𝒗𝒗(𝑡𝑡 + 𝑑𝑑𝑡𝑡) = |𝒗𝒗(𝑡𝑡)| sin𝜃𝜃 �̂�𝑡 + |𝒗𝒗(𝑡𝑡)| cos𝜃𝜃 𝑛𝑛� (S16)

𝜉𝜉(𝑡𝑡) is a δ-correlated Gaussian white noise term, simulating the Brownian force, such that ⟨𝜉𝜉(𝑡𝑡)⟩ =

0, and ⟨𝜉𝜉(𝑡𝑡)𝜉𝜉(𝑠𝑠)⟩ = 2𝛾𝛾𝑘𝑘B𝑇𝑇𝛿𝛿(𝑡𝑡 − 𝑠𝑠), where 𝑘𝑘B is the Boltzmann constant and T is the temperature

of the system (293.15 K),28 in accordance with the dissipation-fluctuation theorem. The potential

𝑈𝑈(𝒓𝒓) is periodic in x with period L (1000 nm), such that 𝑈𝑈(𝑥𝑥) = 𝑈𝑈(𝑥𝑥 + 𝐿𝐿). An asymmetric

periodic potential 𝑈𝑈(𝑥𝑥, 𝑧𝑧 = 0) is applied to the bottom boundary of the simulation box, eq s17,

𝑈𝑈(𝑥𝑥, 𝑧𝑧 = −𝑑𝑑/2 ) = 12𝑎𝑎1(1 + sin �2π𝜋𝜋

𝐿𝐿�) + 1

2𝑎𝑎2(1 + sin �4π𝜋𝜋

𝐿𝐿�); 𝑎𝑎1 = 1, ; 𝑎𝑎2 = 0.25 (S17)

while the top boundary is grounded, 𝑈𝑈(𝑥𝑥, 𝑧𝑧 = 100 𝑛𝑛𝑚𝑚 ) = 0. The potential applied from the

bottom thus decays toward the top, according to the dielectric properties of the medium, creating

the two-dimensional potential landscape 𝑈𝑈(𝒓𝒓). The dielectric of the medium is 2.09. The temporal

waveform 𝑥𝑥(𝑡𝑡) is defined by eq s18, where A is the amplitude of the applied potential, and 𝑓𝑓 is

the frequency of oscillation.

𝑥𝑥(𝑡𝑡) = A ∙ sin(2π𝑓𝑓𝑡𝑡) (S18)

Each particle carriers a charge of +e, where e is the elementary charge. At time t = 0, the

particles are randomly released throughout the simulation area. We propagate the system using the

generalized-α method with a maximum time step size of (2000 x oscillation freq.)-1 for 20

oscillations. The first 4 oscillations are used to equilibrate the system while the final 16 oscillations

are analyzed. We repeat every set of conditions 4 times and report the mean values across all four

replicates. By varying the dynamic viscosity of the solution, 𝜂𝜂, we modify the diffusivity of the

particles, 𝐷𝐷h, according to the Stoke-Einstein equation (𝐷𝐷h = 𝑝𝑝𝑏𝑏𝑇𝑇6𝜋𝜋𝜋𝜋𝜋𝜋

), and the mobility through the

Einstein-Smoluchowski equation, 𝜇𝜇h = 𝑞𝑞𝑆𝑆h𝑝𝑝B𝑇𝑇

. We see a peaked relationship between the distance

traveled by the particles per oscillation (equivalent to our experimental ratchets 𝑞𝑞𝑜𝑜𝑜𝑜𝑜𝑜) and the

ratcheting frequency, Figure S11A. Upon decreasing the viscosity (increasing the

diffusivity/mobility), we observe an increase in the peak ratcheting frequency. For low simulated

mobilities (~7 x 10-5 to 7 x 10-4 cm2 V s-1), we observe a linear relationship between the peak

ratcheting frequency and mobility, Figure S11A,B, whereas at higher simulated mobilities (>7 x

10-4 cm2 V s-1) the frequency response becomes broad and does not shift with mobility. In this

mobility regime, the particles no longer experience overdamped transport and—due to inertial

effects—are less spatially confined by the ratcheting potential, Figure S12.

Figure S11. A) Plots of negative mean distance traveled by the ensemble per oscillation as a function of ratcheting frequency for different particle mobilities (diffusivities) for finite element simulations of charged particles in a solvent. The plotted values are the means of four replicates, where the error bar is the standard deviation from the mean. B) Plot of peak ratcheting frequency vs mobility for the finite element simulations in A where each trace was fit to a Lorentzian function. The error bars represent the uncertainty in peak arising from the Lorentzian fit. C) Plots of negative mean distance traveled by the ensemble per oscillation as a function of ratcheting frequency for different particle mobilities for finite element simulations of charged particles in a solvent. The plotted values are the means of four replicates, where the error bar is the standard deviation of the mean.

Figure S12. Density plots displaying particle trajectories of nanoparticles in solution upon application of a ratcheting potential at the bottom boundary through finite element simulations at different time steps. The color scale is normalized separately for every panel, where darker areas indicate higher particle densities. The red dashed traces correspond to the potential applied at the bottom boundary. We display two periods of the ratcheting potential for two mobilities: 2.7 x 10-4 (A) and 8.5 x 10-4 cm2 V-1 s-1 (B) at the same oscillation frequency (30 kHz). In panel B, we observe inertial effects, where the particles are no longer effectively confined by the ratcheting potential.

Fabrication of Devices with Differing Numbers of FEs. We fabricated devices with differing

numbers of FEs (2,4,6,8 and 16 FEs) to determine the effects of the curvature of the electric

potential generated by the FEs on the ratcheting response (as mentioned in the main text). The

devices all have very similar FE shapes, 400-nm separation between FEs and 3 µm separating the

source (drain) electrode from the first (last) FE. All of the devices exhibited strong artefactual

peaks at ~3-6 kHz that we attribute to capacitance between the source (drain) electrode and the

FEs that was present within the ratchet devices without P3HT. Upon deposition of P3HT, the

position of these peaks remains constant while the magnitude increases. We subtracted the

capacative peak from the data by subtracting the scaled frequency response of the pristine (clean)

devices from the same devices with P3HT, Figure S14. We fit the magnitude by which we scaled

the pristine device response to minimize the current at the peak.

Simulation of Electric Potential. We simulated the electric potential in the devices using the finite-

element software COMSOL Multiphysics 5.4. The Pt finger electrodes are ~50 nm and ~10 nm thick, and

Figure S13. Plot of source-drain current as a function of ratcheting frequency for a ratchet device as fabricated without P3HT (red solid) and with P3HT (black solid). To remove the artefactual capacitive peak at ~6 kHz, we subtracted a scaled frequency response of the clean device (red dashed) to generate the expected frequency response of the P3HT only device (black dashed).

are embedded in a 180 nm SiO2 layer, so that they are separated from the transport layer by 130 nm. The

FEs are spaced 400 nm apart with 3 µm between the first (last) FE and the source (drain) contacts. The

transport layer is 70 nm thick, and both the dielectric and transport layers have a dielectric constant of 2.09.

Above the transport layer we simulate a 3 µm layer of air (dielectric constant of 1), and the potential is set

to 0 V at the top of the air layer, representing the potential at infinity.

Figure S15. Plot of simulated electric fields versus x coordinate at the bottom of the P3HT transport layer during a 10 V sinusoidal oscillation for devices fabricated with 16 (A), 8 (B), 6 (C), 4 (D), and 2 (E) FEs.

Figure S14. Plot of simulated electric fields versus x coordinate at the bottom of the P3HT transport layer (black) and top of the transport layer (blue) during a sinusoidal oscillation.

References

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