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Nonequilibrium 1=f Noise in Rectifying Nanopores
Matthew R. Powell,1 Ivan Vlassiouk,2 Craig Martens,3 and Zuzanna S. Siwy1
1Department of Physics and Astronomy, University of California, Irvine, Irvine, California 92697, USA2Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
3Department of Chemistry, University of California, Irvine, Irvine, California 92697, USA(Received 19 May 2009; published 9 December 2009)
We report a single rectifying conically shaped nanopore system with ion current fluctuations whose 1=f
noise characteristics observed at low frequencies are voltage dependent. Switching the voltage polarity
allows one to switch between a system that produces equilibrium and nonequilibrium 1=f ion current
fluctuations. The nonequilibrium fluctuations in the high-conductance state of the device are characterized
by exponential dependence of the normalized power spectrum on voltage. The asymmetric 1=f noise is
found characteristic for rectifying polymer nanopores and absent in pores with Ohmic current-voltage
curves.
DOI: 10.1103/PhysRevLett.103.248104 PACS numbers: 87.16.dp, 05.40.�a, 72.70.+m
Noise properties of solid-state nanopores have attracteda great deal of interest as a tool for analyzing processesoccurring on the nanoscale as well as a nuisance affectingthe application of nanopores as biosensors [1,2]. Studyingnoise in nanopores is important because fluctuations of ioncurrent in time are the main detection signal in single-molecule analysis and carry information on interactions ofions and the nanopore structure. Ion current through solid-state nanopores often exhibits noise characteristics thatvary from pore to pore, which can make precise andreliable nanopore biosensing difficult. The variationsamongst pores are most pronounced in the low-frequencyrange as seen in the power spectra. In the recent publicationby Smeets et al., two solid-state nanopores with similaropening diameters showed 1=f noise whose power spectradiffered in the low-frequency regime around 1 Hz by 2orders of magnitude [3]. Nanopore noise at higher frequen-cies above the 1=f region has recently been described interms of fluctuations of surface charge density on the porewalls [4]. Whether the 1=f noise in nanopores is also due tosurface or rather bulk effects still remains a topic of debate[5]. Moreover, all nanoporous systems reported thus farexhibited equilibrium fluctuations of current or conduc-tance [3,6], defined as a voltage-independent power spec-trum magnitude normalized by the square of the meancurrent [7]. Ion current has therefore been used as a probefor these equilibrium processes, which occur in the poreeven without applied voltage [6].
In this Letter we present a system consisting of singlerectifying conically shaped nanopores [8,9] whose 1=fnoise can be tuned at will by the external voltage. 1=fnoise of ion currents in the high-conductance state has anonequilibrium character and the magnitude of the nor-malized power spectrum increases with voltage in an ex-ponential manner. In contrast, 1=f fluctuations in the lowconductance state are practically voltage independent. Thisvoltage-tunable 1=f noise is not found in cylindrical,
Ohmic nanoporous systems. Studies with neutral poresindicate that 1=f noise in polymer nanopores does notoriginate from fluctuations of the surface charge densityon the pore walls but rather from dynamic properties of thesolution filling the pore, e.g., fluctuations of ionic diffusioncoefficient and related ionic mobility.The single conical nanopores used here were prepared
by the track-etching technique described previously [10].Briefly, the method entails irradiating 12 �m thick films ofpolyethylene terephtahlate (PET) or polyimide (Kapton50HN, DuPont) with single energetic ions, and subsequentasymmetric etching of the irradiated material [11]. Thefabrication process is known to create carboxyl groupson the pore walls at a density of �1 group per nm2 inboth polymer materials. At neutral and basic pH values,the pore walls are negatively charged, while at acidic pHvalues, the pore walls become neutral. The single conicallyshaped nanopores that we studied in this Letter had anarrow opening between 2 and 8 nm in diameter, and awide opening of 400 nm for PET and 1 �m for polyimidepores. The diameter of the narrow opening, called the tip,was estimated by the electrochemical method describedbefore [8]. PET and polyimide pores are known to differ intheir surface properties: polyimide membranes are signifi-cantly smoother compared to the etched PET surface [11].The noise properties of the single conical nanopores werecompared with single cylindrically shaped nanopores inPET obtained by symmetric etching of the irradiatedsamples in a diluted solution of NaOH at 70 �C.The conical shape together with the surface charge
determines the ion current rectification properties of PETand polyimide nanopores [8,9,12]. When the carboxylgroups are deprotonated, the nanopores are selective forpositively charged ions (e.g., Kþ) with the preferentialdirection of the cation flow from the narrow entrance tothe wide opening [8,9]. Conically shaped nanopores inPET and polyimide show similar rectification properties
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[Fig. 1(a)]. Larger currents for polyimide pores are due tothe larger diameter of the base of the cone. Current-voltagecurves represent average transport properties of the pores,thus in order to study transient characteristics of the currentwe looked in detail at ion current signals in time. Twominute long time series were studied in the voltage rangebetween �1000 mV to þ1000 mV with a 50 mV step(Axopatch 200B, 1322A Digidata, Molecular Devices,Inc) [Fig. 1(b)]. We applied a sampling frequency of10 kHz, and the signals were filtered with 2 kHz Besselfilter. The data were analyzed by looking at the powerspectra in the frequency regime between 0.1 Hz to 1 kHz.
Figure 1(c) shows power spectra of two ion current timeseries recorded for a single PET conical nanopore in 0:1MKCl, pH 8 at �1000 mV and þ1000 mV, along with theexpected level of the shot noise, calculated as Sshot ¼
2ehIi, where hIi is the average value of the current and eis the elementary charge [6]. For all studied recordings, themeasured noise is typically at least an order of magnitudehigher than the shot noise. The power spectra SðfÞ of ioncurrents show power-law dependence 1=f� in the low-frequency range between 0.1 Hz and 100 Hz for�1000 mV. The range of the 1=f noise depends, however,on voltage and becomes larger with increasing voltage. Forvoltages of both polarities between 100 mVand�300 mV,the 1=f� scaling with � close to 1 is observed only overone decade between 0.1 and 1 Hz. The 1=f scaling of thepower spectrum extends to the region up to �10 Hz forboth voltage polarities larger than 300 mV. Currents fornegative voltages above 700 mV show 1=f noise for threedecades up to 100 Hz. For some conical pores, the expo-nent � had a tendency of being lower for positive voltages,as shown in the example in Fig. 1(c).The large difference in the power spectra amplitude for
currents recorded at positive and negative voltages in thelow-frequency range is surprising [Fig. 1(c)]. It is becausethe average current for �1000 mV is only 3 times higherthan the average current forþ1000 mV [see Figs. 1(a) and1(b)], while the magnitude of power spectrum at 1 Hzdiffers by a factor of approximately 100. In order to quan-tify the differences in the power spectra for recordingsperformed at different voltages, we examined the magni-tude, SðfÞ at f ¼ 1 Hz, normalized by the squared value ofthe average current. Figure 1(d) clearly indicates that thecurrents for negative voltages are much noisier than forpositive voltages. There is a voltage range for whichSð1 HzÞ=hIi2 is constant, and for absolute negative volt-ages greater than �400 mV it increases with voltage in aroughly exponential manner [Fig. 1(d)]. We show this forone Kapton and two PET pores but it was observed for allsix examined rectifying nanopores. The noise asymmetrywas stable over many days and was not sensitive to wash-ing the pores with different electrolytes and buffers.Polyimide pores are known to have much smoother wallscompared to PET pores [11]. Both types of pores, however,rectify the current and exhibit the exponential increase ofthe normalized power spectrum with voltage in the high-conductance state. Surface roughness nonetheless influen-ces the magnitude of the normalized power spectrum,which is lower for smoother polyimide pores.All current recordings that we considered were quite
stable, with the standard deviation staying below 8 pA.We noticed that if the current signal through a pore wasmore noisy (standard deviation above 50 pA) the magni-tude of the power spectrum increased up to 100 times andthe 1=f noise occurred over a wider frequency rangecompared to pores with stable ion current signal. Theseobservations are in accordance with the report by Smeetset al. [3] as well as with our previous recordings of 1=fnoise in highly fluctuating sub-2-nm double-conically andconically shaped PET nanopores [13,14]. The hourglasspores were studied only in the low-voltage regime for
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FIG. 1 (color). (a) Current-voltage (I-V) curves of a singlePET conical nanopore with openings of 5 nm and 400 nm (opentriangles), and a single polyimide pore with openings of 2 nmand 1000 nm (open squares) recorded in 0:1M KCl, pH 8, 2 mMPBS buffer. The pores do not rectify at acidic pH; an example ofa recording for the PET pore at pH 3.5, 0:1M KCl, adjusted with0:1M HCl is shown (filled triangles). (b) Examples of ion currentsignals in time at pH 8 for the PET pore whose I-V curves areshown in (a). (c) Power spectra of the data shown in (b): for�1000 mV (blue), and þ1000 mV (red), together with power-law fits; dashed lines indicate the levels of the shot noise forþ1000 mV (red) and�1000 mV (blue). (d) Black: SðfÞ=hIi2 forthree pores: two PET pores with the opening diameter of 5 nm(squares with crosses) and 8 nm (filled squares), and 2 nmpolyimide pore (stars). Solid lines indicate exponential fits ofSð1 HzÞ=hIi2 for negative voltages. Error bars for the 8 nm PETpore were calculated according to a standard error analysistaking into account standard deviation of ion current and ofthe values of Sð1 HzÞ. The latter was estimated as a standarddeviation of values found after dividing each time series into 9segments. In blue: exponent � of the 1=f� scaling of the powerspectra for one PET (filled circles) and one polyimide pore (opencircles), calculated in the range between 0.1 Hz and 2.0 Hz. Theerror for � is up to 20%. All power spectra were corrected for theequilibrium noise [6].
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which the distinct switching between two discrete conduc-tance states were observed [13].
In the recent publication by Hoogerheide et al. [4]fluctuations of the surface charge on walls of single siliconnitride pores were identified as the main source of thewhitenoise in the frequency region 0.1 kHz–10 kHz, whichfollows 1=f noise. In the study with the silicon nitridesystem, current in pores with no surface charges exhibitedlower noise compared to recordings through fully chargedpores. This is because the charged groups on the pore wallsundergo a dynamic protonation/deprotonation reaction,which in turn causes current fluctuations. In order to in-vestigate the effect of the time dependent protonation stateof carboxyl groups on ion current in our polymer nano-pores, we performed current recordings at different pHvalues, at which the pore walls are characterized by differ-ent surface charge densities. At pH 8 the pore walls arefully deprotonated and have charge of�� 1e=nm2, whileat pH 3 the PET and polyimide pores are uncharged. Wealso studied ion current at pH values of 7.0 and 5.5. At allexamined pH values and voltages of both polarities largerthan 300 mV, current fluctuations exhibited 1=f noise overat least two decades of frequency. In contrast with theresults by Hoogerheide et al., lowering surface chargedensity caused an increase of the power spectrum magni-tude (Fig. 2), which we believe is related to the number ofcharge carriers residing in the pore, being significantlylower for acidic pH [15]. Our results thus suggest thatthe surface charge fluctuations are not the dominant factorin the presence and properties of the 1=f noise in rectifyingpolymer pores.
The presence of the surface charge is however respon-sible for the ion current rectification as well as the noiseasymmetry. For neutral PET and polyimide pores, thedependence of the normalized power spectrum SðfÞ=hIi2on voltage shows similar behavior for both voltage polari-ties (Fig. 2). Surface charge can also be screened by highionic concentration of 1M KCl, which eliminated the 1=fnoise asymmetry as well (data not shown).
We therefore attribute the 1=f noise scaling to thedynamic properties of ions filling the pore volume, forexample, fluctuations in the ionic diffusion coefficient. A
similar explanation has been given for 1=f noise observedin the rectifying solid-state device of the Schottky diode[16]. 1=f noise in the Schottky diode as well as othernanopore systems [3,5] is typically described by theHooge formula [17], which relates current fluctuations tothe number of charge carriers Nc via the so-called Hoogeparameter �H:
SðfÞ ¼ I2�H
fNc
: (1)
Our experimental data indicate that there is a range ofthe applied voltage for which the factor SðfÞ=hIi2 is indeedconstant in agreement with Eq. (1). In order to check theapplicability of the Hooge formula to the full examinedvoltage range we looked at the voltage dependence of thenumber of charge carries in the pore. We numericallydetermined the average concentrations of potassium andchloride ions along the pore axis by solving the Poisson-Nernst-Planck equations for a conical pore with homoge-neous surface charges in contact with 0:1M KCl [18].Figure 3 shows the results of the numerical calculations:for the high-conductance state of the device (negativevoltages) the ionic concentrations, and consequently Nc,steeply increase with voltage, while for the low conduc-tance state, the concentrations decrease. When we applythis qualitative relation into Eq. (1) it becomes apparentthat the Hooge formula with voltage-dependent number ofcharge carriers does not describe our data. For negativevoltages, Eq. (1) predicts a decrease of the SðfÞ magnitudeand not an increase, as observed experimentally. Thus, 1=fnoise in the rectifying system of polymer pores cannot bedescribed by the Hooge formula and the 1=f noise in thehigh-conductance state of the device has a nonequilibriumcharacter. Our conical nanopore is indeed the first ionicnanopore system for which nonequilibrium fluctuationshave been observed.The presence of nonequilibrium fluctuations in our pores
is in strong contrast to the recordings obtained previouslyfor silicon nitride pores [3]. Both types of pores, siliconnitride and polymer, contain negative charges on the porewalls but they differ in their transport properties. Silicon
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FIG. 2. Power spectra of ion current through a conical 8 nmPET (a), and 2 nm polyimide (b) nanopore recorded in 0:1M KCland various pH values, as indicated in the figure.
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nitride pores typically behave like Ohmic resistors [3],while conically shaped nanopores with charges on thepore walls rectify the current. We therefore also studiedthe ion current time series characteristics for cylindricallyshaped polymer pores, which are known to produce linearcurrent-voltage curves [19]. Here we examine the proper-ties of ion currents through two single cylindrically shapednanopores in PET with diameters of 20 nm and 100 nm,respectively. Although the diameter of the second poreseems very large, the currents measured were comparableto the values obtained for the 5 nm conically shaped nano-pore. For the cylindrical pores we found the character ofthe power spectrum virtually independent of the voltagepolarity, with the 1=f noise occurring only for frequenciesbetween 0.1 Hz and �2 Hz [Fig. 4(a)]. The factorSð1 HzÞ=hIi2 remained constant in the voltage range be-tween 300 mVand 1000 mV for both polarities, suggestingthe presence of equilibrium current fluctuations [Fig. 4(b)],and applicability of Eq. (1). The experiments with cylin-drical nanopores also suggest that the nonequilibrium typeof ion current fluctuations is intrinsic to polymer, rectifyingnanopores and is not related to our experimental settings.
The presence of nonequilibrium fluctuations in our coni-cal rectifier is surprising. To the best of our knowledge,nonequilibrium fluctuations have not been observed pre-viously in ionic systems, but there are reports of nonequi-librium 1=f noise in solid-state devices [20] explained bylocal heating of a sample or the existence of voltage-dependent energy barriers [7]. In our system, the formercause is unlikely since the total measured currents are low,and the noise properties of the nanopores are stable in timeso that the system is in its steady state. Changes in theapplied voltage, however, cause a significant increase ofionic concentrations in the pore, which might lead to ioniccrowding. As a consequence, regions with various local
resistances might be created, which would also causeenhancement of the 1=f noise [20].In conclusion, we presented a system consisting of a
single conical nanopore in a polymer film, which rectifiesthe current and produces nonequilibrium 1=f current fluc-tuations. We have also found a relationship between theshape of a charged nanopore, the corresponding current-voltage curve, and the type of noise characteristics of ioncurrent. Further studies are necessary to elucidate themechanism behind nonequilibrium 1=f noise.The research was supported by the National Science
Foundation (CMMI 0825661). Z. S. S. received supportfrom the Alfred P. Sloan Foundation. The single ion irradi-ation was performed at the Gesellschaft fuer Schwerionen-forschung, Darmstadt, Germany. Discussions withProfessor James Rutledge are greatly acknowledged.
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FIG. 4 (color). (a) Power spectra of ion current time seriesrecorded for a single PET cylindrical nanopore with a diameterof 100 nm. Recordings were performed in 0:1M KCl, pH 8,þ1000 mV (red) and �1000 mV (blue). The power spectrawere corrected for the equilibrium noise [6]. The dotted lineindicates the shot noise level for both recordings. (b) Magnitudeof power spectra at 1 Hz normalized by the squared value ofaverage current for different voltages. High values of the nor-malized power spectrum at low voltages can most probably beattributed to very low values of the current, which decreased thesignal to noise ratio.
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