NANO LETTERS
Screening in Nanowires and Nanocontacts: Field Emission, Adhesion Force, and Contact Resistance
2009 Vol. 9, No. 12 4306-4310
X.-G. Zhang*,†,‡ and S. T. Pantelides§,| Center for Nanophase Materials Sciences, Computer Science and Mathematics DiVision, and Materials Science and Technology DiVision, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, and Department of Physics and Astronomy, Vanderbilt UniVersity, NashVille, Tennessee 37235 Received August 4, 2009
ABSTRACT The explanations of several nanoscale phenomena such as the field enhancement factor in field emission, the large decay length of the adhesion force between a metallic tip and a surface, and the contact resistance in a nanowire break junction have been elusive. Here we develop an analytical theory of Thomas-Fermi screening in nanoscale structures. We demonstrate that nanoscale dimensions give rise to an effective screening length that depends on the geometry and physical boundary conditions. The above phenomena are shown to be manifestations of the effective screening length.
A metallic object screens an external electric field so that, in the absence of a current, the internal electric field approaches zero within a very short distance from the surface. This distance is called the screening length. Similarly, the Coulomb potential of a charged impurity in a metal is screened completely within a screening length. In macroscopic objects, Thomas-Fermi (TF) theory provides a good description of the screening length, which we call the TF screening length and denote it by λTF. Typical λTF values in metals are smaller than 1 Å. When there is a current, the internal electric field is proportional to the resistivity of the metal. In a macroscopic crystalline metal that is free of defects and impurities, whose resistivity approaches zero at low temperatures, screening again makes the internal field go to zero, even in the presence of a current. In contrast, in an undoped semiconductor or insulator at low temperatures, the internal electrostatic potential drops linearly with distance. Doped semiconductors, on the other hand, especially degenerate semiconductors, exhibit metal-like screening with λTF that can be quite large: nanometers or even micrometers. The issue of screening in metallic or metal-like nanoobjects, i.e., objects with at least one nanoscale dimension, * To whom correspondence should be addressed. † Center for Nanophase Materials Sciences, Oak Ridge National Laboratory. ‡ Computer Science and Mathematics Division, Oak Ridge National Laboratory. § Materials Science and Technology Division, Oak Ridge National Laboratory. | Vanderbilt University. 10.1021/nl902533n CCC: $40.75 Published on Web 10/21/2009
2009 American Chemical Society
has so far received limited attention. Two papers1,2 considered generic cylindrical systems (metallic, molecular, etc.) with a wide range of values for the ratio R/λTF, where R is the radius of the cylinder. They applied the TF approximation to show that, if the nanowire cross section is comparable to λTF of the pertinent bulk material, screening is incomplete, i.e, the effective screening length can be much larger than the bulk λTF value. When the nanowire’s cross section is smaller than the bulk λTF and its length is on the order of 10λTF to 20λTF, the nanowire can sustain an insulator-like linear voltage drop along its length. Broader consequences of nanoscale screening have not been explored. In this Letter we show that screening in nanoscale objects has broader consequences, and use the results to explain a variety of phenomena that have already been observed but have eluded fundamental understanding. More specifically, we first obtain analytic solutions of the Poisson equation in the TF approximation for several different boundary conditions. We show that nanoscale screening is not merely an accumulation of a charge density on the surface that decays with λTF, as in a macroscopic metal. Instead, the actual effective screening length can be either longer or shorter than λTF of the pertinent bulk material, depending on the geometry and the physical setup. We then give three specific examples where nanoscale screening has manifest consequences. The first example is the field enhancement factor β of a nanowire in field emission. The existing theory finds a factor significantly larger than the measured values.3 Screening from adjacent emitters is widely accepted as the source of
discrepancy, but no quantitative model is available.4 We obtain a simple analytic formula for field emission by a single nanowire that explains the experimental data. The key point is that, along the length of the wire the screening length is effectively infinite, leading to a much smaller β. In the second example, the attractive force between two metal surfaces, the so-called adhesion force, was suggested5 to decrease exponentially over the distance of λTF, but the measurement of a metal tip and a surface6 shows that the decay length is much greater than λTF. Applying our solution, we show that the actual decay length depends on the sharpness of the tip, thus resolving the discrepancy between theory and experiment. In the third example, a contact resistance ranging from 150 to 490 Ω was observed in Au break-junction experiments,7-11 but its origin is unexplained. We show that it is due to the finite screening length on the surface of the electrodes. However, in this case, the effective screening length is only about half of the λTF as a result of the lateral variation of the charge distribution. We first consider the general case of an isolated piece of metal placed in an external field described by the electrostatic potential Vext(r). In the absence of an electric current, the electron Fermi energy (or chemical potential) EF in the metal is a constant. To be consistent with the TF model, we measure the changes in the electrostatic potential due to an applied field from those of the energy of a particular atomic core state,12 which removes the oscillations of the electrostatic potential on the atomic scale, as one would obtain from a first-principles calculation. The Poisson equation can be written in the form1 (see Supporting Information for detailed derivation) ∇2δV(r) -
1 λTF2
δV(r) ) 0
(1)
where δV is the total perturbative potential, including both Vext and the induced potential, the TF screening length is defined as λTF ) 1/4πe2D(EF), and D(EF) is the electron density of states at the Fermi energy. Equation 1 is applicable to a metal of any shape and size. However, the effect of finite screening length becomes significant when the size of the metal is nanoscale. For a cylindrical solid nanorod of length L along the z direction and radius R, the general solution using the cylindrical symmetry is δV(r, z) )
∫
∞
-∞
(
dλ′F(λ′)ez/λ′J0 r
1 1 2 λ λTF2
)
(2)
where J0 is the zeroth-order Bessel function, which is oscillatory in r for λ′ < λTF. For λ′ > λTF, J0 has an imaginary argument and grows exponentially with r. The coefficients F(λ′) are to be determined by the boundary conditions. Equation 2 is the central result of this paper. We illustrate the evaluation of F(λ′) for the case of a uniform electric field along the z direction, Vext ) -Eextz, where Eext is the strength of the electric field, for two well-known limiting boundary Nano Lett., Vol. 9, No. 12, 2009
conditions, and later apply the scheme to realistic cases. We consider a cylinder of length L and radius R. In the first limiting case, we take R f ∞ while keeping L to nanometer size, resulting in a nanothickness flat disk. In this limit there should be no dependence of the electrostatic potential on the radial coordinate r, which is only possible if F(λ′) ) A1[δ(λTF + λ′) - δ(λTF - λ′)] where A1 is a constant. The radial part is J0(0) ) 1, and we have δV(r, z) ) A1(e-z/λTF ez/λTF) for a flat disk. The second limiting case is a nanowire with L f ∞ and a nanoradius R. In this limit the electric field along the nanowire is constant, corresponding to an electrostatic potential of the form -A2z at r ) 0. To satisfy this requirement, we find F(λ′) ) (1/2)A2 limλ1f∞ λ1[δ(λ1 + λ′) - δ(λ1 - λ′)]. We now simplify the general solution given in eq 2 by approximating it as a linear combination of the two limiting cases: δV(r, z) ≈ A1(e-z/λTF - ez/λTF) - A2zI0(r/λTF)
(3)
where I0(r/λTF) ) J0(ir/λTF) is the modified Bessel function of zeroth order. The ratio between A1 and A2 is to be determined by the boundary conditions. It is easy to see that, if R . L (a flat disk), then A2 f 0, and only the first term in eq 3 contributes. Likewise, if L . R (long wire), then A1 f 0, and only the second term contributes. Previous numerical solutions of eq 11,2 are for the case of a nanowire connected to two electrodes at both ends. For this case, we give our analytic solution in the Supporting Information and show that our solution covers all the cases studied numerically in previous works.2 One problem not discussed in previous works is the case of an isolated (disconnected from any electrodes) nanowire in a uniform electric field. In this case, we can express the solution in terms of an effective permittivity similar to that of a dielectric material. We define the effective permittivity as ε ) EextL/∆V, where ∆V is the potential difference between the two ends of the nanowire and EextL is the potential drop over the same distance in vacuum. Here we list two limits. The first is when R . L, i.e., a flat disk: εdisc )
L 2λTF tanh(L/2λTF)
(4)
The second is the opposite limit of L . R: εthinwire ) 1 +
( ) ()
R L R I ln λTF 0 λTF R
(5)
A similar expression is obtained for a nanotube of length L and radius R: εnanotube ) 1 + 2e2n1(EF) ln(L/R)
(6)
where n1(EF) ) 2πD(EF)RδR is the one-dimensional density of states, and δR is the nominal thickness of the nanotube. The above three equations tell us that the permittivity of a 4307
Figure 1. Models for field emission. (a) In the sphere model, the nanowire is approximated by a sphere of radius R, and the opposite electrode is approximated by a concentric sphere a distance d away. (b) The nanorod model used in this paper.
metal is always finite. It simply becomes very large in a macroscopic sample where L . λTF for a disk or R . λTF for a wire. These results can be compared to dielectric nanowires studied previously.13 So far we have discussed the cases of a nanowire attached to electrodes at both ends, and of an isolated nanowire not attached to any electrodes. An important application of nanowires, field emission, involves nanowires attached to an electrode at only one end. As an application of our theory to this geometry, we calculate the field enhancement factor of a single nanowire attached to one electrode. This is an important parameter for studying field emission. Previous theory4 considers only the charge at the tip of the nanowire, and approximates the charge distribution with a sphere of radius R, as illustrated in Figure 1a. The resulting enhancement factor β ) d/R, where d is the separation of the electrodes, is too large compared to the experiment. An additional adjustable parameter s is then introduced to obtain β ) sd/R, but no formal derivation of s has been given. We use a more realistic model, shown in Figure 1b, and apply the solution eq 2. For L . R, the dominant term is still the λ′ f ∞ term, but because there is no longer any symmetry along the rod, the zeroth order term is not canceled and yields δV(r, z) ≈ A0I0(r/λTF) inside the nanowire. The voltage drop from the tip of the nanowire to the unattached electrode is the applied voltage ∆V, which determines A0. We find, for the limit L . R . λTF and assuming a spherical tip, β ) Ez)L/2d/∆V ) 2.414d/[R ln(2L/R)], and for the limit L . λTF . R, β ) 2d/L (see Supporting Information for derivation). For both limits, the normalized field enhancement factor βR/d is only a function of the aspect ratio L/R. Contrary to the conventional belief, in both limits the normalized field enhancement factor βR/d decreases with the increasing aspect ratio L/R. In Figure 2 we compare experimental data for nine different materials3,14-28 with our theory as well as the sphere model4 β ) d/R. Most of the experimental data fall within the boundary of the two limiting curves. A few experimental data points that are higher than the λTF ) 0 curve are probably due to the actual nanowire tips being sharper than the spherical tip assumed in our formula. The decrease of βR/d with increasing L/R is clearly seen in experimental data. We note that, although λTF may vary a great deal from sample to sample, the general trend that βR/d decreases with L/R is the consequence that the 4308
Figure 2. Normalized field enhancement factor, βR/d, as a function of the aspect ratio L/R for nanowires. The experimental data are as follows: Co: ];14 Ni: right 4;15 Mo: left 4;16 ZnO: O,3 0,17 9,18 +,19 4,20 2,21 3,22 /;23 ZnO/Ga: right 2;24 CuO: 1;25 AlN: ×;26 SnO 2: b;27 NiSi 2: (.28 Theory is βR/d ) 2.414/ln(2L/R) for λTF ) 0 and βR/d ) 2R/L for λTF ) ∞. The sphere model is βR/d ) 1.
effective screening length along the nanowire, λ′, is essentially infinite. The second example that the effective screening length can be much greater than λTF is given by the screening effect on the attractive force between a metal tip and a metal surface. In an experiment6 between a W tip and a Au surface, the decay distance of this adhesion force was found to be 2 Å, 4 times the value of λTF ) 0.5 Å for W. Although the sharpness of the tip was suspected to be the cause of the disagreement, no theoretical model can satisfactorily explain this. When two metal plates are brought close to each other, surface charges accumulate due to the Fermi energy difference between the plates. Neglecting the exchange interaction, the Coulomb energy between these charges is the interaction energy between the two plates and produces an adhesive force. This energy is approximately given as (see Supporting Information) E)
(λ12 + λ22)(∆E)2 -2d/(λ1+λ2) e e(λ1 + λ2)
(7)
where λ1 and λ2 are the effective screening lengths of the two metal plates, ∆E is the difference in the Fermi energy between the two plates, and d is the separation of the two surfaces. Indeed, the energy scales with d with a decay length of (λ1 + λ2)/2, as suggested in ref 5. We wish to find an approximate exponential form of the screening charge near the metal tip. Let us suppose that the tip has an aspect ratio of R, i.e., the radius of the tip is R ) Rz at a distance z away from the apex. There is no closed form solution, so we solve the problem numerically. The numerical solution is shown in Figure 3. For a very sharp tip with R ) 0.07, the effective decay length along the z-direction is 8λTF, which is needed to obtain a 4-fold increase in the decay length after averaging over the screening lengths of the flat surface and the tip. This is clearly a geometric effect, and the measured decay length depends strongly on the sharpness of the tip. Finally, we consider the problem of contact resistance due to incomplete screening at the electrode-nanowire contacts. In many break junction measurements of quantized conducNano Lett., Vol. 9, No. 12, 2009
inside one of the electrodes, z < -L/2 or z > L/2, where the two electrodes are placed at (L/2, respectively, similar to eq 2, we have δV(r, z) ) V-∞ Figure 3. Numerical solution of the electrostatic potential near a sharp metallic tip with λTF ) 0.5 Å and R ) 0.07 under a 1 V applied voltage. The decay length of the electric field along the axis of the tip is 4 Å, 8 times the bulk TF screening length.
tance,7-11 it was necessary to subtract a contact resistance of a few hundred ohms in order to match the measured conductance quantum to G0 ) 2e2/h. The source of this contact resistance has remained a puzzle. For a ballistic nanowire, the electric field inside the nanowire adds a perturbative term in the Hamiltonian, which does not cause dissipation and does not change the quantized conductance. The question is, in order to apply a voltage over a nanoscale distance, how much of that voltage falls inside the electrodes due to the finite screening length in the electrodes? This is essentially the same problem described above and in the Supporting Information for the electrostatic energy between two metal plates, but now with the separation d between two surfaces replaced by the length of a nanowire L. Setting λ1 ) λ2 ) λTF, ∆E/e ) ∆V, where ∆V is the applied voltage, and assuming L . λTF, the charge density is rewritten as F1 ) ∆V/4πλTFL. The voltage drop on each electrode is V1 ) 4πλTF F1 ) λTF∆V/L 2
(8)
The total voltage drop inside both electrodes is 2λTF∆V/ L. This is equivalent to an extra contact resistance of 2λTF/ GL, where G is the conductance of the nanowire. In the absence of scattering in the nanowire, the conductance is proportional to the cross section πR2. Assuming that the volume is conserved during the pulling of the scanning tunneling microscope (STM) tip from the surface, i.e., πR2L ) const, then GL is a constant when the nanowire is pulled during a break junction experiment. Let us now compare this result to the experiments of Au break junctions7-11 in which it was found that the measured quantized conductance is always slightly smaller than G0 ) 2e2/h. Contact resistances of 150,7 315,8 380,9 390,10 or 490 Ω11 were found depending on the experimental conditions. Since the resistance correction is a constant, we can estimate this term at any point of the pulling process. The length L at the last conductance plateau is 20 Å. For Au, the TF screening length is about λTF ≈ 0.6 Å. These values yield a contact resistance of (h/2e2)(2λTF/L) ≈ 774 Ω. This is close to the experimental values considering the simplicity of the model. The most likely reason that the estimate is too large is the neglect of the lateral variation of the screening charge at the contacts, which can effectively reduce the screening length along the direction of the current. We give an analysis below on how this could change the screening length. We consider the electrostatic potential along the surface of a disk-shaped electrode with the contact at r ) 0. For z Nano Lett., Vol. 9, No. 12, 2009
∫
∞
0
(
dλ′F(λ′)e-(|z|-L/2)/λ′J0 r
)
1 1 2 λ λTF2 (9)
where V-∞ - V+∞ ) ∆V is the applied voltage. When the nanowire radius is close to the Fermi wavelength, we cannot find the charge variation along r without solving the wave functions of the electrons. However, we note that the lateral variation should be dominated by the Fermi wave vector. For example, if the point where the nanowire makes the connection to the electrode is viewed as a point defect potential on the electrode surface, then the charge response on the electrode surface is a charge oscillation29 with a wave vector 2kF similar to the Friedel oscillations, where kF is the Fermi wave vector of the electrode. In general, we consider that the charge in the electrode near the nanowire contact has a radial distribution of J0(2kFr). This means that we need to consider λ′ )
λTF
√1 + 4kF2λTF2
(10)
as the dominant contribution in eq 9. For Au, the Fermi wavelength is about 2π/kF ≈ 5.2 Å. The screening length in the electrodes given by eq 10 is λ′ ≈ 0.34 Å, which is much shorter than λTF. The contact resistance calculated using the new screening length is (h/2e2)(2λ′/L) ≈ 439 Ω. The actual solution can be approximated as a combination of a uniform surface charge plus an oscillatory charge with a wave vector 2kF. Thus the contact resistance predicted by the TF screening model should be between 439 and 774 Ω. This range overlaps the range of measured values. The large variation among the experimental values may be due to the different STM tip materials used and the differences in the lengths of the nanowires. These factors are not considered in our estimate. In summary, we obtained an analytic solution for charge screening in metallic and semiconducting nanowires and nanocontacts. The solution can be useful in many applications involving nanoscale structures. Here we gave examples in field emission, adhesive force, and contact resistance. The result for field emission, that βR/d decreases with L/R, is unexpected but makes sense if one considers the sphere model for which βR/d ) 1 is the upper limit for the field enhancement factor. Thus the maximum enhancement for fixed R and d occurs when L ≈ R, which indeed is what our model predicts. This result suggests that increasing the aspect ratio of nanowires may not be the best optimization approach for field emission. Acknowledgment. This research was conducted at the CNMS sponsored at ORNL by the Division of Scientific User Facilities, U.S. DOE. The work was further supported by 4309
the DOE Office of BES, Division of Materials Science and Engineering, Grant No. DEFG0203ER46096, and by the McMinn Endowment at Vanderbilt University. Supporting Information Available: Detailed solutions for the nanowire problem and comparison with previous works. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Liang, G. C.; Ghosh, A. W.; Paulsson, M.; Datta, S. Electrostatic potential profiles of molecular conductors. Phys. ReV. B 2004, 69, 115302. (2) Nitzan, A.; Galperin, M.; Ingold, G.-L.; Grabert, H. On the electrostatic potential profile in biased molecular wires. J. Chem. Phys. 2002, 117, 10837–10841. (3) Zhang, Q.; Zhang, H. Z.; Zhu, Y. W.; Feng, S. Q.; Sun, X. C.; Xu, J.; Yu, D. P. Morphological effects on the field emission of ZnO nanorod arrays. Appl. Phys. Lett. 2005, 86, 203115. (4) Flip, V.; Nicolaescu, D.; Tanemura, M.; Okuyama, F. Modeling the electron field emission from carbon nanotube films. Ultramicroscopy 2001, 89, 39–49. (5) Rose, J. H.; Ferrante, J.; Smith, J. R. Universal binding energy curves for metals and bimetallic interfaces. Phys. ReV. Lett. 1981, 47, 675– 678. (6) Schirmeisen, A.; Cross, G.; Stalder, A.; Gru¨tter, P.; Du¨rig, U. Metallic adhesion and tunnelling at the atomic scale. New J. Phys. 2000, 2, 29.129.10. (7) Brandbyge, M.; Schiotz, J.; Sorensen, M. R.; Stoltze, P.; Jacobsen, K. W.; Norskov, J. K.; Olesen, L.; Laegsgaard, E.; Stensgaard, I.; Besenbacher, F. Quantized conductance in atom-sized wires between two metals. Phys. ReV. B 1995, 52, 8499–8514. (8) Muller, J.; Krans, J. M.; Todorov, T. N.; Reed, M. A. Quantization effects in the conductance of metallic contacts at room temperature. Phys. ReV. B 1996, 53, 1022–1025. (9) Hansen, K.; Laegsgaard, E.; Stensgaard, I.; Besenbacher, F. Quantized conductance in relays. Phys. ReV. B 1997, 56, 2208–2220. (10) Costa-Kra¨mer, J. L.; Garcı´a, N.; Olin, H. Conductance quantization histograms of gold nanowires at 4 K. Phys. ReV. B 1997, 55, 12910– 12913. (11) Garcı´a, N.; Costa-Kra¨mer, J. L. Quantum-level phenomena in nanowires. Europhys. News 1996, 27, 89–91. (12) Payne, M. C. Electrostatic and electrochemical potentials in quantum transport. J. Phys.: Condens. Matter 1989, 1, 4931–4938. (13) Mahan, G. D. Nanoscale dielectric constants: Dipole summations and the dielectric function in nanowires and quantum dots of cubic materials. Phys. ReV. B 2006, 74, 033407. (14) Vila, L.; Vincent, P.; Dauginet-De Pra, L.; Pirio, G.; Minoux, E.; Gangloff, L.; Demoustier-Champagne, S.; Sarazin, N.; Ferain, E.; Legras, R.; Piraux, L.; Legagneux, P. Growth and field-emission
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NL902533N
Nano Lett., Vol. 9, No. 12, 2009
