Pauli Repulsion-Induced Expansion and Electromechanical

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Pauli Repulsion-Induced Expansion and Electromechanical Properties of Graphene Hui Wang, Xiaonan Shan, Hong-Yuan Chen, and Nongjian Tao Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b03955 • Publication Date (Web): 14 Dec 2016 Downloaded from http://pubs.acs.org on December 14, 2016

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Pauli Repulsion-Induced Expansion and Electromechanical Properties of Graphene Hui Wang,1 Xiaonan Shan,2* Hong-Yuan Chen,1* and Nongjian Tao1,2* 1

State Key Laboratory of Analytical Chemistry for Life Science, School of Chemistry and

Chemical Engineering, Nanjing University, Nanjing 210093, China 2

Center for Bioelectronics and Biosensors, Biodesign Institute, Arizona State University, Tempe,

AZ 85287, USA

Corresponding Authors *E-mail: [email protected], [email protected], [email protected];

ACS Paragon Plus Environment

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Because graphene has nearly zero density of states at the Dirac point, charging it must overcome Pauli repulsion. We show here that this repulsion causes graphene to expand, which is measureable with an optical edge-tracking method despite that graphene is the strongest material. The expansion increases quadratically with applied voltage as predicted by theory, and has a coefficient of ~10-4 per V at 1 V. Graphene has many attractive properties, but it lacks piezoelectricity, which limits its electromechanical applications. The observed Pauli repulsioninduced expansion provides an alternative way to electrically control graphene dimension. It also provides a simple and direct method to measure the elastic properties of graphene and other low dimensional materials.

Keywords: Pauli repulsion, graphene expansion, quantum capacitance, electromechanical properties of graphene, Young’s modulus of graphene.


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Charging an object requires energy to overcome electrostatic repulsion, which is described by electrostatic capacitance. This principle has been used in electromechanical actuators1. Unlike most materials, graphene has nearly zero density of states at the Dirac point 2-4. Consequently, charging of graphene, i.e., adding electrons or holes to graphene, requires energy to overcome Pauli repulsion described by quantum capacitance. The quantum capacitance of graphene dominates its total capacitance, so the Pauli exclusion effect is much greater than the electrostatic repulsion in graphene. Here we show that Pauli repulsion leads to expansion of graphene, which is measureable with an optical method despite that graphene is the strongest material. Graphene is non-piezoelectric

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, and this effect provides an alternative way to

electrically control the dimension of graphene, which is useful for developing electromechanical applications with graphene 9-14. We measure the Pauli repulsion-induced expansion as a function of charge and graphene size, and describe the results with a thermodynamic model. We further demonstrate that this effect can be used to determine the Young’s modulus of graphene. The free energy (F) of a graphene sheet on the substrate can be expressed as a sum of surface energy, elastic energy and charging energy, by

= −σA+

− + ,

(1)

where σ is surface energy per unit area, E is the Young’s modulus, t is the thickness of graphene, A0 and A are the areas of graphene before and after charging, Q is charge, and U(Q) is the charging energy. In graphene, U(Q) is dominated by Pauli repulsion energy. The charging energy can be expressed in terms of capacitance. In the case of graphene, the capacitance is primarily due to quantum capacitance, , which is originated from Pauli repulsion. The graphene quantum capacitance is not a constant, and it depends on the voltage (or


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charge), which has been described theoretically

15-17

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. The actual quantum capacitance of

graphene deviates from the theoretical prediction due to defects and impurities

17-21

, so we

measure it for the samples studied in the present work (see Supporting Information). Using the measured quantum capacitance (C(V)) and minimizing the free energy (Eq. 1) with respect to A, the Pauli repulsion-induced strain is expressed as (see Supporting Information) ∆

=

,

(2)

where L is the length and ∆L is the change of length in the direction of measurement, and c(V) is graphene quantum capacitance per unit area. This equation predicts that charging of graphene leads to its expansion, and by measuring the expansion, we can determine the Young’s modulus of graphene.


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Figure 1. Pauli repulsion induces graphene expansion. (a) Experimental setup to determine the Pauli repulsion-induced graphene expansion with an optical edge-tracking method. (b) Energy diagrams and Pauli repulsion-induced expansion of graphene at different bias voltages. (c) Phase contrast image of a monolyaer graphene edge, where the black arrow marks the edge displacement caused by lateral translation of the sample stage. (d) Intensity profiles across the edge before and after the lateral translation, where blue dots are experimental data, and red lines are fitting to a polynomial function.

To charge graphene we applied a sinusoidal voltage to a monolayer graphene sample on a gold-coated glass substrate with respect to a reference electrode in 0.1 M sodium fluoride (NaF) solution (Figure 1a, and also Supplementary Information). The applied potential introduces extra electrons or holes to the graphene, which experience Pauli repulsion because of the low density of states near the Dirac point, leading to the expansion of graphene (Figure 1b). To accurately measure the Pauli repulsion-induced graphene expansion, we developed an optical edge-tracking method to measure the edge displacement of graphene associated with charging. The method used a phase contrast microscope with a 40x objective to image the monolayer graphene. Figure 1c is a typical phase contrast image of the graphene, which reveals an edge of the sample. The intensity profile across the edge within a region of interest (ROI) is shown as dots in Figure 1d. This intensity profile or image contrast at the edge of graphene comes from the phase difference between the single layer graphene and gold substrate. We fitted the intensity profile with a polynomial function (red line, Figure 1d). When the graphene expands, the intensity profile shifts laterally. By fitting the intensity profile with the function with the lateral shift as an adjustable parameter, we determined the lateral shift as a function of time and voltage.


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Figure 2. Optical edge-tracking method: calibration and detection limit analysis. (a) Comparison of the edge displacement determined by the optical edge-tracking method (red), and actual displacement by the piezoelectric translation stage (black). (b) FFT spectrum of the edge displacement determined with the optical edge-tracking method (red) and background noise obtained in the absence of the translation (blue). (c) Linear relationship between the edge displacements determined with the optical edge-tracking method, and the actual displacement.

To validate and calibrate the edge-tracking algorithm, we translated the graphene sample laterally by applying a periodic voltage with frequency 10 Hz to the piezoelectric translation stage of the microscope, and measured the associated lateral edge movement of graphene by the optical edge-tracking method (red line, Figure 2a). The results obtained with the optical edgetracking method are in excellent agreement with the actual displacement (black line, Figure 2a). Fast Fourier Transform (FFT) of the graphene edge movement reveals a pronounced peak at 10 Hz (red curve, Figure 2b). The amplitude of the peak at 10 Hz is proportional to the maximum displacement of the graphene edge. The detection limit of the method was determined by the background noise, which was measured by turning off the microscope translation stage (blue line, Figure 2b). From the background noise at 10 Hz in the FFT spectrum, we found that the detection limit (standard deviation) was about 0.8 nm. The displacement determined from the optical edge-tracking method is a linear function of the actual displacement determined from the


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piezoelectric translation stage over a wide range (Figure 2c). Using this empirical calibration curve, we determined the graphene expansion.

Figure 3. Pauli repulsion induces expansion in a monolayer graphene. (a) Optical image of a monolayer graphene with a length of 174 µm in the direction marked with a yellow arrow. The two red dashed lines mark two opposite edges where displacements are tracked. (b) Phase contrast image of the region marked by a red box in (a). (c) Raman spectrum of the graphene sample where the ratio of G to 2D peaks is ~1:2, indicating monolayer graphene. (d) AFM of the region marked with a red box in (a). (e) Edge


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displacement determined with the optical edge-tracking method (blue dots), and fitting to a sinusoidal function with frequency twice of that of the applied voltage (black solid line). The applied voltage is shown as red dashed line. Note that the location of the edge is marked by a yellow dashed line in (b). (f) FFT spectrum of the graphene edge displacement in response to a 5 Hz voltage modulation showing a major peak at 10 Hz, and a minor peak at 5 Hz.

After establishing the optical edge-tracking method, we applied it to measure the Pauli repulsion-induced graphene expansion by applying a periodic voltage modulation to the monolayer graphene. To confirm that the graphene sample was monolayer, we imaged the sample with optical contrast (Figures 3a, b) and Atomic Force Microscopes (AFM) (Figure 3d), and performed Raman spectroscopy (Figure 3c). In response to the applied voltage modulation at 5 Hz (red dashed line), we observed a periodic expansion (blue dots) of the graphene edge (yellow dashed line in Figure 3a) with frequency at 10 Hz (Figure 3e). The observed frequency doubling is expected because the expansion is a quadratic function of voltage (Eq. 3). This frequency doubling is more clearly shown in the FFT spectrum of the edge displacement (Figure 3f). From the amplitude of the second harmonic peak in Figure 3f, we determined that the graphene expanded by ~8.8 nm for a voltage modulation of 0.65 V. We note that in each sample we tracked the displacements at two opposite edges (e.g., red dashed lines in Figure 3a), and took the sum of the displacement magnitudes as the graphene expansion (see also Figures S2). The length of the monolayer graphene in the direction of expansion was 174.5 µm (Figure 4a), from which we found that the expansion coefficient (strain) is 5 x 10-5 per V at 0.65 V. We did not observed hysteresis, buckling or delamination of graphene during the experiment.


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To further examine that the observed graphene expansion was due to relative motion between graphene and substrate, we tracked the displacements of defects on the substrate. In contrast to the graphene edges, we did not detect any potential-induced displacements in the defects (Figures S3 and S4). We also tracked the edge of the gold substrate (on glass), and did not observe any detectable expansion in the gold film during potential modulation (Figure S5). These control experiments show that the measured potential-induced expansion arises from graphene rather than the substrate. The second harmonic peak in the FFT spectrum is expected from the quadratic dependence of the expansion on voltage, but the FFT spectrum also shows a smaller harmonic peak at 5 Hz (in Figure 3f). This harmonic peak is due to a small offset charge in the graphene, which shifts the Dirac point to Vpzc, the potential of zero charge. Consequently, V in Eq. 2 is replaced with V-Vpzc, and square of it leads to both quadratic and linear terms, giving rise to harmonic and second harmonic peaks in the FFT spectrum. As shown in Supplementary Information, the Pauli repulsion-induced expansion in graphene in the direction of measurement can be further simplified as,

∆| =

!

(3)

where ∆L|2f is the second harmonic displacement, L0 is the length of graphene in the direction of measurement, and V is the amplitude of voltage modulation. Figure 4a plots the second harmonic edge displacement of monolayer graphene samples with different lengths vs. V, and fitting of the experimental data with Eq. 3. The good agreement between the measured data and the fitting further validates the prediction of the voltage dependence of the expansion by the theory. Eq. 3 also predicts the length dependence of the expansion. To examine this prediction, we measured


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the expansion of graphene samples with different sizes. The results as plotted in Figure 4b confirm that the linear dependence of the Pauli repulsion-induced graphene expansion with the sample lengths. The systematic length dependence also further confirms that the measured expansion is an intrinsic property of graphene, rather than a sample specific property.

Figure 4. Voltage and length dependence, and Young’s modulus measurement. (a) Dependence of graphene expansion on applied voltage for graphene samples with different lengths (101, 174 and 329 µm), and fitting of the data (solid lines) with Eq. 3. (b) Dependence of graphene expansion on length at different voltages (different colors), and fitting of the data (solid lines) with Eq. 3. (c) Measured quantum capacitance of a monolayer graphene vs. voltage. (d) Histogram of monolayer graphene Young’s modulus obtained from 36 data points with different samples (9 samples) and conditions (four different potentials: 0.55 V, 0.60 V, 0.65 V, and 0.7 V).

Fitting the voltage and length dependent expansion with Eq. 3 yields values for !

=

∆|"# " $%%

. Since t = 0.35 nm for monolayer graphene, and c0, the quantum capacitance at zero


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voltage, is straightforward to measure with the same sample (Supporting Information), we can determine E, the Young’s modulus of graphene, from the values. We measured the quantum capacitance of the graphene samples under the same condition as we measured the charginginduced expansion (Supporting Information). Figure 4c shows the quantum capacitance of a monolayer graphene on gold electrode vs. the voltage modulation amplitude. The value and shape deviate the theoretical prediction of quantum capacitance for ideal graphene, but agree well with those reported in literature

17, 18, 22

. Using the quantum capacitance, we found that the

average Young’s modulus is ~ 0.4 TPa with a standard deviation of 0.1 TPa (Figure 4d). This Young’s modulus is smaller than that measured with AFM (E = 1.0 ± 0.2 TPa) for pristine graphene

23-25

. We attribute the smaller Young’s modulus to microstructured defects and grain

boundary ripples26, 27 in the graphene samples prepared with chemical vapor deposition (CVD) method. Recent works have shown that these defects in graphene samples can significantly reduce the Young’s modulus 28-30, and the Young’s modulus has been found to be in the range of 0.25 - 0.82 TPa for chemically derived monolayer graphene

28, 31

. For this reason, we refer the

Young’s modulus as effective Young’s modulus. The graphene samples described above were on gold surface, and the surface adhesion between the graphene and substrate may affect the accuracy of the Young’s modulus determined based on the graphene expansion 32. Graphene surface adhesion was measured to be 0.45 J/m2 by Bunch et al

33, 34

. The surface area change of graphene measured in the present work is as small

as 5 × 10 m2, which leads to a surface adhesion energy change of 2.25 × 10 J. Compared to the associated change in the elastic energy, this surface energy change is 3 orders of magnitude smaller, indicating that the surface adhesion effect is insignificant. We have tracked the individual edge expansion on both sides of graphene, and observed similar expansion


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amplitudes, which is consistent with the weak adhesion between the graphene samples and the substrate (Figure S2). To further examine effect of graphene-substrate adhesion on the measured Young’s modulus, we performed the measurement by placing graphene on hydrophobic and hydrophilic surfaces created by modifying gold electrodes with 1-hexanethiol and cysteamine, respectively. Figure 5a compares the expansion of monolayer graphene on bare gold, cysteamine- and 1-hexanethiol-modified gold surfaces. To determine the Young’s modulus of graphene on these surfaces, we measured quantum capacitance at the Dirac point for each surface (see Supporting Information). We measured expansion of graphene samples on bare gold, cysteamine- and 1-hexanethiol-modified gold surfaces and the extracted average Young’s moduli were 0.35 ± 0.1, 0.46 ± 0.1 and 0.34 ± 0.1 TPa, respectively, for graphene on the different surfaces (Figure 5b). This result shows that surface adhesion does not significantly affect the Young’s modulus extracted from the Pauli repulsion-induced graphene expansion.

Figure 5. Pauli repulsion-induced graphene expansion and Young’s modulus measurement of graphene on different surfaces. (a) Graphene expansion vs. the applied voltage on bare gold (black), hydrophilic cysteamine (blue)- and hydrophobic 1-hexanethiol (green)-Au surfaces. (b) Effective Young’s modulus of graphene on different surfaces. We measured 6 samples on bare gold, 4 samples on cysteamine- and 4 samples on 1-hexanethiol-modified gold surfaces.


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AFM indentation has been used to measure the Young’s modulus of single layer graphene, and establishes graphene as the strongest material

35-37

. The AFM method requires

suspension of a graphene sheet over an open hole, pushing the graphene with the AFM probe, and calculating the Young’s modulus with a mechanical model

38, 39

. Measuring the Young’s

modulus of graphene based on the Pauli repulsion-induced expansion is direct and simple as it does not require these sample preparation steps, and can measure multiple samples with arbitrary shapes and sizes, simultaneously. Furthermore, it can determine the Young’s modulus in different directions, which is important for anisotropic materials. We examined the Young’s modulus of monolayer graphene in random directions, and detected no difference within the experimental error, which is consistent with the six-fold symmetry of graphene 40. We anticipate that the method can determine anisotropic mechanical properties in materials with lower symmetries. Finally, we point out that Eq. 3 can be re-expressed as ∆|"#

=

! ,-

,

(4)

where Vol is the volume of the material, which shows that charging-induced expansion is proportional the surface-to-volume ratio of the sample. For this reason, we expect that this charge-induced expansion can be observable in other low dimensional materials. When applying Eq. 4 to other materials, both Pauli and Coulomb repulsion effects may be included in the capacitance. Eq. 4 also explains why this phenomenon has not been previously observed in bulk materials. In conclusion, we have observed Pauli repulsion-induced expansion graphene for the first time with a sensitive optical edge-tracking method. The effect leads to electromechanical control


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of graphene dimension, despite that graphene is non-piezoelectric, which could be employed for electrochemical actuation of graphene or other low dimensional materials. The measured electromechanical expansion increases quadratically with voltage as predicted by theory, and reaches ~10-4 per V at 1 V. The effect provides a direct and simple method to measure the mechanical properties, such as Young’s modulus, of nanomaterials. The measured results for graphene are in good agreement with those by AFM indentation. Unlike the AFM method, the present method does not require fabrications of suspended graphene structures, can measure the Young’s modulus in different directions, and multiple samples with arbitrary shapes and sizes, simultaneously. We anticipate that similar effect (either Pauli or Coulomb repulsion) occurs in other low dimensional materials.


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ASSOCIATE CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: Details of sample preparation, measurement methods, equations of charging-induced graphene expansion, measurement of graphene quantum capacitance, and voltage modulation (PDF)

AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected], [email protected], [email protected]; Author Contributions H.W. carried out the experiment and analyzed data. X.S. and H.Y.C. supervised the experiment. N.T. conceived the idea and wrote the paper. Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS. We thank Drs. Kangping Chen and David Ferry for stimulating discussions, and CNSF (#2137008 and #2137902) and Gordon and Betty Moore Foundation for financial support.


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Pauli Repulsion-Induced Expansion and Electromechanical

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