Electronic Transport on the Nanoscale: Ballistic Transmission and Ohm s Law

Transcription

1 Electronic Transport on the Nanoscale: Ballistic Transmission and Ohm s Law NANO LETTERS 2009 Vol. 9, No J. Homoth, M. Wenderoth, T. Druga, L. Winking, R. G. Ulbrich, C. A. Bobisch, B. Weyers, A. Bannani, E. Zubkov, A. M. Bernhart, M. R. Kaspers, and R. Möller*, 4. Physikalisches Institut, UniVersity of Göttingen, Göttingen, Germany, and Department of Physics and Center of Nano Integration Duisburg-Essen (CeNIDE), UniVersity of Duisburg-Essen, Duisburg, Germany Received December 15, 2008; Revised Manuscript Received February 2, 2009 ABSTRACT If a current of electrons flows through a normal conductor (in contrast to a superconductor), it is impeded by local scattering at defects as well as phonon scattering. Both effects contribute to the voltage drop observed for a macroscopic complex system as described by Ohm s law. Although this concept is well established, it has not yet been measured around individual defects on the atomic scale. We have measured the voltage drop at a monatomic step in real space by restricting the current to a surface layer. For the Si(111)-( 3 3)-Ag surface a monotonous transition with a width below 1 nm was found. A numerical analysis of the data maps the current flow through the complex network and the interplay between defect-free terraces and monatomic steps. As the size of the metallic interconnects in integrated circuits is decreasing, the fundamental processes limiting the electric conduction on the nanometer scale need to be understood completely. Local deviations of a perfect crystalline structure lead to the scattering of the conduction electrons which causes the loss of directed momentum and eventually the dissipation of energy. This not only limits the operation of the circuit but also is the starting point for electromigration which finally leads to the failure of the device. Macroscopically the dissipated power is equal to the product of electric current and voltage. The latter is given by the difference of the electrochemical potential (µ ec ). The goal of the present paper is to evaluate the variation of µ ec in the vicinity of an elementary defect with high spatial resolution extending into the quantum mechanical regime. The effect of a pointlike scatterer on the ballistic transport of electrons in a conductor was described by Landauer in the framework of classical and quantum mechanical theory. 1,2 He predicted the local electric dipole. However, its experimental observation is lacking up to now, mainly because µ ec is not directly accessible for a three-dimensional conductor in an experiment. The key feature of this work is the study of the electric current in a two-dimensional electron gas (2DEG) restricted to the surface layer. This enables full local access to µ ec by scanning tunneling potentiometry (STP). The analysis of the data provides (i) the local electric field as the gradient of µ ec, (ii) the current density, and (iii) University of Göttingen. University of Duisburg-Essen. the local conductivity at each point in the surface. In addition, the experiment allows study of the spatial variation of the electrochemical potential around individual defects. For locally different directions of the flux of electrons this reveals significant variations of the microscopic conductivity. The information about the local processes is used in a bottom up approach to analyze the transition from the atomic to the mesoscopic scale. We connect here the local conductivities, e.g., of the barrier of a step edge, with the global conductance of an ensemble of terraces, which is measured in more macroscopic experiments, e.g., four-probe measurements. 3-7 STP introduced by Muralt and Pohl 8 used the tip of a scanning tunneling microscope (STM) to measure simultaneously the topography of the sample and the local electrochemical potential µ ec. The latter is evaluated by adjusting the external bias voltage such that the average tunneling current between tip and sample vanishes. The electrochemical potential of the tip, which balances the electronic distribution of the sample, will be referred to as µ ec,stp (x,y) in the following. This method was used to map the variations of µ ec around structural defects 9-16 in metallic and semiconducting samples. For thin metal films it was shown that grain boundary scattering is the largest contribution to the global resistivity. 11,17 Analyzing the electric transport in thin films of bismuth, Briner et al. 14 succeeded in evaluating the electrochemical potential around a nanometer-sized rectangular hole /nl803783g CCC: $40.75 Published on Web 03/11/ American Chemical Society

2 Our study is performed for the well-defined, pure twodimensional electronic system: the silver-induced ( 3 3) superstructure on the Si(111) surface. It is formed by covering the surface of a (111)-oriented silicon single crystal with one monolayer of silver ( atoms/cm 2 ). As shown by photoelectron spectroscopy and scanning tunneling spectroscopy, the reconstruction yields a partially occupied electronic surface state providing a sufficient conductivity directly accessible at the surface. 4 Electrons in this state behave as a two-dimensional free electron gas 18 (2DEG) with an effective mass of 0.13m e. Especially on n-type silicon charge transport through the 2DEG is electrically isolated from the bulk states of the Si by a space charge layer. 5 The experiments were performed in situ in ultrahigh vacuum. 11,19-21 The samples were prepared from n-type (phosphorus) Si(111) wafers with a specific resistance of 130 Ω cm. For the first set of experiments, wafers with a miscut of 0.5 (type I) were chosen, leading to rather parallel steps with a density of about 30 steps/µm. The lateral current was applied by two gold tips of a three-probe scanning tunneling microscope, 22 which were brought into contact with two metallic silver islands in close vicinity ( µm). The handling was controlled by means of a scanning electron microscope (SEM) with a resolution better than 100 nm. Prior to the presented experiments, the macroscopic resistance was evaluated as a function of the contact probe spacing confirming transport within a two-dimensional system. Figure 1a displays the geometric structure of an area of µm 2. Most of the steps correspond to monatomic steps. About one-third of them correspond to double steps and a few to 3-fold steps. The terraces in between the steps are atomically flat. Panels b and c of Figure 1 show that µ ec,stp (x,y) measured simultaneously to the data of Figure 1a exhibits only a minor monotonous gradient constant on the terraces and varies abruptly at the step edges. Figure 1d shows cross sections of the topography and µ ec,stp data at a particular step. The variation of µ ec,stp is monotonous and occurs within 6 Å (20% to 80%). The transition is displaced relative to the step by about 12 Å. It is well-known that the electronic surface state is reflected by a monatomic step leading to a pattern of standing waves in the electronic density of states. This has been also observed for the Si(111)-( 3 3)-Ag surface by scanning tunneling spectroscopy. 24 However, there are no indications of oscillations in µ ec,stp, on neither the upper nor the lower terrace. The observation of constant values of µ ec,stp on the terraces and abrupt variation at the step proves that the current transport is restricted to a 2D layer at the surface and is not much bypassed via the silicon substrate. Probing µ ec,stp at the same step for different current densities as well as for the opposite direction of the current reveals a linear response. It should be noted that the dissipated power per area is no longer given by σ(3µ ec,stp ) 2 because the voltage drop is due to quantum mechanical reflection and the thermalization of the carriers occurs on a much longer length scale. Figure 1. Scanning tunneling microscopy and potentiometry of the Si(111)-( 3 3)-Ag surface. The area is µm 2. The insets in (b) and (e) show scanning electron microscope images for an area of µm 2. Two gold tips, to the left (top) and to the right (bottom), are in contact to silver islands. The tip for the tunneling experiments can be seen in the middle. (a) The topographic image, the gray scale covers about 1 nm. The circles denote domain boundaries of the reconstruction, which are barely visible in the topography. A zoom of the domain boundary at the lower right is displayed in the inset in the upper right corner. Type A domain boundaries are marked by dashed, type B by full circles. (b) The electrochemical potential simultaneously measured at an average current density of j extern = 0.4 A/m perpendicular to the steps. The result of the corresponding simulation based on a network of Ohmic resistors is displayed in (c). (d) Line scans across the edge of a monatomic step displaying the tip height in constant conductivity mode (blue and green) and the effective electrochemical potential (red and black). To improve the signal-to-noise ratio, a step edge with rather large signal was chosen, the lateral current was increased (j ) 2 A/m) and the signal was averaged over several lines. Sections scanned upward and downward are presented demonstrating the absence of artifacts. (e) The electrochemical potential simultaneously measured at an average current density of j extern = 0.26 A/m parallel to the steps. The result of the corresponding simulation is displayed in (f). To analyze the relation between the electrochemical potential and the current density at any position of the surface, an iterative numerical procedure was applied to the data, using the equations div jb ) 0 and Ohm s law jb ) σ3µ ec,stp (x,y). The first guarantees the continuity of the Nano Lett., Vol. 9, No. 4,

3 Table 1 sample I current perpendicular to the steps sample I current parallel to the steps sample II isotropic step distribution σ 2DEG,10-6 Ω -1 /0 > ( ( 1100 single steps, Ω -1 m ( 500 < ( 600 multiple steps, Ω -1 m ( 300 <40 domain boundary A, Ω -1 m ( ( 5 domain boundary B, Ω -1 m -1 > ( 500 current, and the latter yields the local relation between the current density jb(x,y), the conductivity σ(x,y,jb(x,y)), and the electrochemical potential µ ec,stp (x,y). On a discrete grid, the problem is equivalent to a network of resistors. To match the STM resolution, up to 10 5 nodes are included. On the basis of the topography, which was recorded simultaneously with µ ec,stp (x,y), any point is attributed to either the plane surface or a atomic step, each having a constant specific conductivity within the frame. A close look on the data reveals two types of domain boundaries on the reconstructed terraces. According to Sato et al. 23 type A (dashed circles in Figure 1) is tentatively assigned to Ib, while type B (full circles in Figure 1) is assigned to Ia. For a complete description the corresponding conductivities have to be taken into account. Using the boundary condition given by µ ec,stp (x,y) at the four sides of the frame and the external current density, the problem can be solved analytically, yielding µ ec,stp (x,y) at any point inside. Iteratively, the conductivities are varied until the best-fit between calculated and measured µ ec,stp (x,y) data is achieved. Panels c and f of Figure 1 illustrate the calculated potential distributions for the data of panels b and e of Figure 1 based on the specific conductivities for the step edge and the twodimensional electron gas (2DEG) listed in Table 1. The excellent agreement between the calculation and the measured µ ec,stp (x,y) data is evident. It should be noted that the simulated electrochemical potential only depends on the relative conductivities. To obtain the absolute values the average current densities of the simulation and the experiment are matched. The latter may be calculated knowing the total current and the lateral distribution of the average current density, which is determined in arbitrary units from the gradient of the electrochemical potential measured at various points between the electrodes. To understand the geometric effects in more detail, a second type of sample (type II) with minimal miscut angle was studied. The samples (n-type 130 Ω cm) with dimensions of mm 3 (length widths thickness) were electrically contacted by molybdenum foils 2 mm apart from the imaged regions. During formation of the Si(111)-( 3 3)-Ag superstructure, the excess Si atoms diffuse on the surface and form the topmost layer. In the case of an almost perfectly flat surface, this leads to a self-assembled two-layer system with two atomic levels Figure 2a displays the surface structure as well as the electrochemical potential measured for an area of 1 1 µm. The measured height is used for the height of the pseudo-3d representation, and the electrochemical potential is displayed by color coding. Figure 2b illustrates the calculated potential distribution for the data of Figure 2a based on the specific conductivities for the step edge and the free terrace given in Table 1. The current density corresponding to these data is plotted in Figure 2c; the magnitude is color coded. A vector field plot for a small area is given in the inset. On comparison of the data of panels b and e of Figure 1, it is rather surprising that the average of the conductance given by the ratio between the current density and the difference in the potential is only about 30% higher if the current flows parallel to the steps. Since the number of steps to overcome is lower by more than a factor of 10, a much larger conductivity is expected. However, this is not the case, as can be simply verified by comparing the macroscopic conductances observed by the total current between the electrodes for the two different geometries. As a consequence, as listed in Table 1 the effective conductivities of the step edges for different prevalent directions of the current are rather different. This may be explained by the transmission through a tunneling barrier as a function of the angle of incidence. The transmission can be calculated analytically for plane waves impinging on a rectangular barrier. It is maximal if the angle of incidence is perpendicular to the step edge (ϑ ) 0). It remains up to 90% of the maximum for ϑ < 30, but it drops to less than 1% for ϑ > 70. Ifthe macroscopic direction of the current is perpendicular to the steps, the majority of the electrons will also locally arrive with small ϑ. Hence, the transmission will be close to its maximum value. This explains the good agreement between the simulation and the measurement although only one value Figure 2. (a) Topography and µ ec,stp (x,y) data (1 1 µm 2 )of sample II. Since the height varies only between two atomic layers, the height information was set to the value of either the higher or the lower level, to help the visualization of the data. The color scale represents the measured µ ec,stp (x,y) data varying by 1.5 mv from blue to red. (b) Calculated potential distribution and (c) calculated current density of the same area. The arrows in the inset indicate the current directions Nano Lett., Vol. 9, No. 4, 2009

4 for the conductivity of the step is used, independent of the local angle of incidence. If the transport occurs preferentially parallel to the steps, the transmission decreases by more than a factor of 10. For sample II the geometric structure is more or less isotropic and the observed step conductance is given by the average over different transmission coefficients. However, this will be dominated by the large transmission at small angles of incidence, leading to a value, which in principle should be a little bit smaller than the one for the current perpendicular to the steps. The best fit to the data yields a value which is even a little bit higher. However, -1 both values lie in the range of σ Step ) 5000 ( 2500 Ω-1 m given by Matsuda et al. 24 determined by a more macroscopic four-probe method and by the reflectional phase shift of Friedel oscillations. The conductivity on the free terraces, hence the free twodimensional electron gas (2DEG) of the surface state, is best evaluated using the data of sample II, which averages over different structure sizes and different current directions. For sample I only a lower limit can be given if the current is perpendicular to the steps. For the parallel direction the current is restricted to narrow channels due to the very low transmission of the step edges. We attribute the reduction of the apparent conductivity by a factor of 10 to the diffuse scattering of the electrons by the corrugated step edges. As the mean free path of the electrons for inelastic scattering of 70 nm (applying the Drude-Sommerfeld theory, using 28 E f ) 250 mev, m* ) 0.13m e, and the 2DEG conductance of 3080 µs/0, the mean free path is evaluated to 70 nm) is larger than the width of the terraces, the effective mean free path is reduced in analogy to the reduction of the sheet conductivity due to imperfections of the surface discussed by Fuchs and Sondheimer. 29 The value of about 3000 µs/0 for the 2DEG is up 1 order of magnitude larger than those reported for four-probe methods. This is not surprising since in more macroscopic measurement the role of the steps cannot be neglected. The image of the current density, e.g., in Figure 2c, shows that the current percolates the structure minimizing the number of steps because of the low transmission across the steps. The conductance determined by four-probe methods is thus given by the conductance of the percolation network rather than the one of the 2DEG alone. On the basis of the conductivities derived above, the macroscopic conductance can be calculated for four-probe measurements on different length scales. Figure 3 illustrates the results using the geometric structure of large scale STM images. The apparent sheet conductance decreases with increasing sheet size and finally saturates. Their variance decreases with sheet size as well. This behavior reflects that local effects average on larger scales. The length scale of 500 nm at which saturation of the sheet conductance occurs is given by the microscopic structure of the percolation network. The corresponding conductivity of approximately 800 µs/0 is close to the conductance determined by four-probe measurements with 8 µm probe spacing. 30 In a recent experiment with only 1 µm probe spacing, a conductivity of 1500 µs/0 was determined on n-doped substrate. 5 On p-doped material, a Figure 3. Calculation of the macroscopically observable sheet conductance as function of the sample size: An STM topography of 2 2 µm 2 (a) and the specific conductivities from Table 1 were used to calculate the effect of percolation. The blue (red) dots represent the horizontal (vertical) direction. The black line displays the mean value of both. maximum conductance of 3000 µs/0 was reported. 6 However, as shown recently 31 this is probably due to the additional bulk contribution, as for p-type Si the current may penetrate deep into the bulk. Our experiment gives access to the spatial variations of the electrochemical potential within an electric conductor with nanometer resolution. At a step edge the local electrochemical potential exhibits a rather localized monotonous transition. The major variation occurs in a very short range of about 6 Å. Our measurements cover the transition from the microscopic transport processes, i.e., quantum mechanical tunneling at step edges and diffusive scattering in the twodimensional electron gas, to the macroscopic resistance. This finally bridges the gap between the elementary processes and the electric resistance observed in macroscopic experiments. Although the transport is ballistic on the nanometer scale, a correct numerical description of the electrochemical potential is obtained using the model of diffusive transport. This gives the unique opportunity to evaluate the specific conductivities of the defect-free surface and across atomic steps. Moreover, the numerical simulation yields the direction and magnitude of the local current density at any point with nanometer resolution. Measurements with the prevalent direction of the current either perpendicular or parallel to the step edges demonstrate the limitation of using specific conductivities. The transmission depends on the angle of impinging charge carriers and may vary by more than 1 order of magnitude. Furthermore, if the steps restrict the current in the two-dimensional electron gas to long and narrow channels, the apparent conductivity of the free surface is reduced, as the effective mean free path decreases due to diffuse scattering at the roughness of the step edges. In conclusion the conductivities which have to be attributed to the free terrace and the steps depend on the geometric structure on a length scale of the electron s mean free path. Nevertheless, the analysis by a network of Ohmic resistors provides a correct numerical description of the observed electrochemical potential for the different experimental situations. Our data explain how the results of previous Nano Lett., Vol. 9, No. 4,

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