Imaging Electron Flow

Transcription

1 Imaging Electron Flow New scanning proe techniques provide fascinating glimpses into the detailed ehavior of semiconductor devices in the quantum regime. Mark A. Topinka, Roert M. Westervelt, and Eric J. Heller Semiconductor heterostructures have revolutionized solid-state physics and its applications. Most of us use the fruits of this revolution every day in CD and DVD recorders and players, cellular telephones, laser-ased telecommunications, satellite television, and much more. The technology, ased on atomic layer-y-layer growth using molecular eam epitaxy (MBE), is sophisticated, remarkale, and marketale. One class of semiconductor heterostructures, the twodimensional electron gas (DEG), has een a focal point for theorists and experimentalists, and a wellspring of new physics. A DEG can e produced at low temperatures at an interface of two distinct layers (a so-called heterojunction) doped neary with atoms that donate electrons. The electrons at such a junction are confined to the lowest quantum state in the direction normal to the interface; y charging gate electrodes on the top surface of the heterostructure some distance away to repel them, the electrons can e further confined in the other directions to make dots, wires, resonators, and other shapes. That technology has led to celerated discoveries including the in- Mark Topinka, now a Uranek Postdoctoral Fellow in the applied physics department at Stanford University, received his PhD from Harvard University under Bo Westervelt, Mallinckrodt Professor of Applied Physics and of Physics. Rick Heller is a professor of chemistry and physics at Harvard. teger and fractional quantum Hall effect (QHE), 1 the Coulom lockade and single-electron transistors, and conductance quantization in quantum point contacts (QPCs). The potential for exploiting these and many other quantum effects is spawning new fields of single electronics 3 and spintronics 4 new approaches to logic that use single electron charges and spins to represent its of data and the new area of quantum information processing, ased on the coherent interaction of quantum mechanical quits. 5 Despite all the eautiful experiments already performed on DEGs and all that is riding on the new science and phenomena made possile y them, researchers have een lind until recently as to how electrons actually move through them. Most of the knowledge of electron flow in DEGs is indirect, ased on electron-transport measurements of macroscopically averaged quantities. To e sure, many of the statistical properties are known, such as the electron mean free path. But macroscopically averaged parameters do not reveal the details of the fascinating ehavior to e found on the nanoscale. For that, imaging is needed. Imaging a system is essential to understanding its fundamental properties and developing new electronic and magnetic devices. Imagine the difficulty of designing and faricating an integrated circuit from a silicon crystal without the use of an optical or electron microscope. As device sizes continue to decrease, quantum ehavior ecomes important and offers new research and application opportunities. To understand the fundamental ehavior of electrons in this quantum regime and to make functioning devices ased on this ehavior, one must develop ways to visualize the flow of electron charges and spins through semiconductors. The invention of the scanning tunneling microscope (STM) allowed researchers to directly view the pattern of atoms on a material s surface. Additional methods are needed to image the flow of electrons eneath the surface. Otaining images of DEGs inside semiconductors is no easy matter, ecause the electrons are uried eneath the surface and ecause the samples must e cooled to low temperatures to show quantum ehavior. Nonetheless, a numer of groups have recently developed liquidhelium cooled scanning proe microscopes (SPM) for this purpose. Making a two-dimensional electron gas Figure 1 illustrates how a DEG can e created inside a gallium arsenide aluminum gallium arsenide heterostructure. During growth y MBE, atomic layers are added to the heterostructure at a rate of aout one layer per second. Conduction-and profiles can e engineered during growth y changing the Al fraction. Switching aruptly from GaAs to AlGaAs creates a very clean, sharp arrier that keeps electrons inside the GaAs layer. Silicon atoms in the AlGaAs layer act as donors: The electrons ionized from the Si fall over the arrier and are trapped at the GaAs AlGaAs interface, where they can move freely for long distances. At low temperatures, movement in the z-direction (normal to the interface) is frozen out, ut the electrons, trapped in a D flatland sheet, are free to move and interact in the x- and y-directions. Two-dimensional electron gases possess a unique comination of parameters that together make an ideal laoratory in which to explore the fascinating and often surprising ehavior of such systems. The electron s de Broglie wavelength at the Fermi energy is unusually long, typically nm, and it can e an appreciale fraction of the size of a typical device. (Electron wavelengths in metals, y comparison, are typically well under 1 nm.) The Fermi wavelength can e tuned y changing the electron density with a planar gate electrode located eneath the DEG. Because they do not collide often with other parti- 003 American Institute of Physics, S Decemer 003 Physics Today 47

2 Figure 1. A two-dimensional electron gas formed at the interface etween gallium arsenide and aluminum gallium arsenide in a semiconductor heterostructure. The AlGaAs layer (green) contains a layer (purple) of silicon donor atoms (dark lue). Electrons from the donor layer fall into the GaAs layer (pink) to form a DEG (lue) at the interface. The ionized Si donors (red) create a potential landscape for the electron gas; the resulting smallangle scattering smoothly ends electron trajectories, as shown. cles, the electrons have a long phase-coherence length that is, they can travel coherently for many microns as quantum mechanical waves with a well-defined phase and with the same energy. They also have a long mean free path: Electrons can flow through a DEG for tens or even hundreds of microns efore losing track of their initial direction. The motion of electrons in a GaAs AlGaAs DEG is limited y small-angle scattering. Positively charged donor ions create a umpy electrostatic potential, shown in figure 1, that leads to smooth variations in the density of the electron gas and the Fermi velocity. The hills and valleys in the potential landscape are typically much smaller than the Fermi energy. Like light traveling through glass with a smoothly ut randomly varying index of refraction, electrons travel through the DEG along smoothly ent paths. Small-angle scattering is an elastic process, ecause the Si ions that reflect the electrons do not recoil, and the electron waves preserve their quantum coherence even though they steadily change direction as they are uffeted this way and that. The distance over which they lose memory of their initial direction is called the mean free path. Imaging the quantum Hall regime The QHE profoundly changes the way electrons move through a DEG.1 In a strong magnetic field applied perpendicular to the DEG, the electrons no longer travel as free plane waves ut instead occupy a series of discrete Landau levels separated in energy y \wc, the energy associated with the cyclotron frequency wc (\ is Planck s constant). Both the numer of states in each Landau level and the spacing etween the levels scale with the applied field. Motion along the field lines cannot occur, ecause the electron gas is two-dimensional. At a magnetic field for which the quantum states in the highest occupied Landau level are half filled, the familiar classical Hall resistance and nonzero longitudinal resistance are oserved. Quantum Hall plateaus occur at magnetic fields for which the highest occupied Landau level is almost completely filled. The numer of filled Landau levels is called the filling factor n. On a quantum Hall plateau, electrons in the middle of the sample form an incompressile liquid (see PHYSICS TODAY, August 003, page 38). The longitudinal resistance of the sample goes to zero, and the transverse Hall resist48 Decemer 003 Physics Today ance is quantized on plateaus of height (1/n)h/e with integer values of the filling factor n, where e is the electron charge. All of the current through the sample is carried y edge states that pass around its circumference. These edge states correspond to classical skipping orits that repeatedly hit the edge as the electron tries to move in a circle in the magnetic field. The fractional QHE, which produces plateaus in Hall resistance at values (1/n)h/e for certain fractional numers, such as 1/3, is also associated with edge states, ut its source is the correlated motion of electrons in the DEG. The QHE is associated with remarkale spatial structures in the DEG. The structures include predicted spatially striped phases of the quantum Hall liquid that have een investigated using macroscopic measurements. Imaging the properties of a DEG in a strong magnetic field at low temperatures is a particularly useful way to understand the QHE, and a numer of groups have developed cooled scanning proe microscopes for this purpose. Raymond Ashoori s group at MIT has developed a way to image electron flow in the quantum Hall regime using a susurface charge accumulation (SCA) proe.6 An STM tip held aove the surface capacitively couples to the DEG immediately elow. When a small AC voltage is applied etween the tip and the DEG, the resulting flow of charge in the gas induces an oscillating image charge on the tip; that oscillation is detected y a sensitive electrometer. Adding a positive DC voltage to the tip allows spatial profiling of the DEG y creating a small ule composed of a few electron charges eneath the tip. The ule is surrounded y an insulating ring of incompressile fluid in the quantum Hall regime, and it forms an electrically isolated quantum dot that holds a fixed, discrete numer of electron charges.7 Figure is an image of the SCA signal otained as the ule is moved through the DEG y scanning the STM tip aove the surface. As the numer of electrons in the ule changes, the signal intensity oscillates, which results in right strips that form a contour map of the random electrostatic potential inside the quantum Hall liquid.

3 SURFACE POTENTIAL CHANGE Figure. Susurface charge accumulation in a two-dimensional electron gas maps the electrostatic potential experienced y the DEG in the quantum Hall regime with a filling factor n 1. A positive voltage on the tip of a scanning tunneling microscope aove the sample pulls in electrons to create a few-electron ule in the DEG. The closed contours in this.5.5-mm image are caused y the quantization of electronic charge inside the ule: The contours, which arise as individual electrons move in and out of the ule, surround high and low regions of the random electrostatic potential. (Adapted from ref. 7.) a This image directly demonstrates how, thanks to the DEG s incompressiility, the QHE can lock the flow of electrons inside the electron gas. Amir Yacoy (now at the Weizmann Institute of Science) and his colleagues at Lucent Technologies Bell Las developed an alternate way to image the charge of an electron gas. 8 They successfully faricated a single-electron transistor (SET) on the tip of a glass fier and used it as a scanning electrometer proe. The SET was sensitive enough to detect changes in the DEG potential and density variations corresponding to tiny fractions of a single electron. With their proe, they imaged edge states that pass around the circumference of the electron gas in the quantum Hall regime (see figure 3a) The image was produced y measuring the charge induced on the SET electrometer when a voltage was applied to a gate electrode eneath the DEG. The incompressile strips of electron gas next to the edge states allowed the electric field to pass through the DEG to reach the SET; the result was the right strips in figure 3a. The SET tip was also used to image the Hall potential, as shown in figure 3 for a magnetic field near the n = plateau. The Hall potential appears at the edges of the DEG, ut no longitudinal potential is seen. These results show how edge states occur in the quantum Hall regime and directly confirm earlier theoretical predictions. The scanning electrometer tip was also used at Bell Las 9 to image, in a manner similar to the Ashoori group, localized DEG electron states in the quantum Hall regime. The quantum Hall regime imaging y the MIT and Bell Las teams as well as y Paul McEuen 10 at the University of California, Berkeley and Cornell University, y Jürgen Weis and Klaus von Klitzing 11 at the Max Planck Institute for Solid State Research in Stuttgart, Germany, and y Klaus Ensslin 1 at ETH Zürich represents groundreaking achievements that have enriched theories of the QHE and revealed intriguing new effects. Imaging electron flow in low magnetic fields Although many insights have come from imaging the quantum Hall regime, the majority of semiconductor devices operate without an applied magnetic field, making it important to image in that regime, too. Recent imaging +.5 mv.5 mv Figure 3. Imaging with a single-electron transistor. (a) Edge states form along the oundaries of a two-dimensional electron gas in the quantum Hall regime. The right strips in this mm image are incompressile regions of the electron gas next to the edge state for a filling factor n. The incompressile regions allow the electric field from an electrode eneath the DEG to reach the SET. (Negative voltage applied to the two lack gate electrodes has closed the constriction etween them.) () The SET proe can also detect changes in potential. A current flowing through an open aperture etween the electrodes generates a transverse Hall potential that appears along the edges of the electrodes. No longitudinal potential is apparent at the top and ottom of this 7 7-mm image. (Adapted from ref. 8.) y our group at Harvard University, y McEuen, y Charles Smith and David Ritchie at the University of Camridge, and y Ensslin has focused on electron flow patterns in a DEG with no applied magnetic field or with small magnetic fields. Mark Eriksson, working with two of us (Topinka and Westervelt), led the charge in 1996 y directly imaging the mean free path in a DEG for electrons passing through a wide constriction. 13 Four years later, Rolf Crook and colleagues at Camridge imaged cyclotron orits in a DEG at 4. K and interpreted features of their images in terms of the deflections in electron trajectories caused y donor atom density fluctuations and impurities. 14 Using a charged atomic-force microscope tip Decemer 003 Physics Today 49

4 a CONDUCTANCE ( e / h) GATE VOLTAGE (mv) c d e 00 nm 00 nm 00 nm Figure 4. Electron flow through a quantum point contact. (a) Scheme for imaging current flow through a QPC using scanning proe microscopy. Two gate electrodes (yellow) create a narrow constriction in the underlying two-dimensional electron gas. A charged tip (green) depletes the electron gas elow it, creating a divot (red spot) that scatters incoming electron waves, as shown in the simulations (lue). () The conductance of the QPC, measured at 1.7 K, increases in quantized steps as the gate voltage (and QPC width) is increased. The insets elow each step show simulations of the spatial pattern of electron flow for the transverse modes that contriute to the conductance. (c e) Experimental images of electron flow at 1.7 K (left and right) and theoretical simulations (center) for the first three transverse modes of a QPC. The oserved interference fringes spaced y half the Fermi wavelength demonstrate the coherence of electron flow. Because the additional flow, appearing as the QPC ecomes wider, is due to the newly opened-up mode, the image for each transverse mode could e otained y sutracting the raw images from the next lower step. (Adapted from ref. 15.) to end the trajectories of electrons traveling etween two QPCs, they achieved glimpses into the spatial details of electron flow in DEG nanostructures. Those early experiments set the scene for more recent high-resolution images that revealed surprising and important details aout electron flow in DEGs. At Harvard, we used scanning proe microscopy to image the coherent flow of electron waves through a DEG with no applied magnetic field. 15,16 We focused on the pattern of electron flow through one of the most fundamental and widely used nanostructures: a QPC 17, a narrow constriction whose width is comparale to the electrons Fermi wavelength l F (see PHYSICS TODAY, July 1996, page ). As its width is increased, the conductance of a QPC increases in steps of height e /h ecause the electrons travel through individual transverse modes, each of which contriutes e /h to the total conductance. Figure 4a illustrates our technique for imaging the flow of electron waves in a GaAs GaAlAs DEG. Gates on the surface form a QPC whose width could e adjusted y changing the gate voltage. A charged SPM tip capacitively couples to the electron gas elow; for negative tip-to-gas voltages, it can deplete a small, round divot in the DEG that reflects electron waves arriving from the QPC. The pattern of electron waves scattered y the divot under the tip is shown y theoretical simulations in the figure. Some of the electrons reflected y the divot return along their incoming path and travel ack through the QPC, measuraly reducing its conductance. Electrons scattered at other angles have little effect on the conductance ecause they remain on the same side of the QPC. The change in conductance induced y the tip is proportional to the flux of electrons hitting the divot under the tip. As the tip is scanned over the sample, the QPC conductance images the 1 mm Figure 5. Electron flow through a two-dimensional electron gas from a quantum point contact on the first conductance step. The image shows surprisingly narrow ranches that are produced y small-angle scattering from charged donor atoms in the donor layer, as shown in figure 1. The interference fringes, demonstrating quantum mechanical coherence, extend throughout the image. The arrow points to a cusp produced y the focusing effect of a neary impurity atom. (Adapted from ref. 16.) 50 Decemer 003 Physics Today

5 a c Figure 6. Simulations of electron flow. (a) Parallel electron trajectories, going from left to right, form a V-shaped cusp due to focusing y a potential-energy dip caused y a charged donor atom (not seen) aove a two-dimensional electron gas. () A realistic DEG simulation that includes many ionized donors forms several generations of cusps. The electrons travel here from upper left to lower right. (c) Ray-tracing simulations of electron flux emerging from a small opening into a region of random potential reflect the features seen in experimental images of DEG quantum point contact samples. The potential is shown green in the valleys and white on the peaks. The electron flux is coded y height and color, with lue corresponding to regions of low flux; still lower flux is transparent. The shadow of the flux on the potential plot shows where the flux lies relative to the hills and valleys; no guiding valleys are seen. A slight change of the position of the opening changes the location and direction of the ranches. (S. E. J. Shaw, PhD thesis, Harvard University, 00.) electron flux that was there efore the tip was present. With this technique, we could image the patterns of electron flow through the individual transverse modes of the QPC that are responsile for the conductance steps shown in figure Figures 4c e compare experimental images of the flow of electron waves through the first three modes of the QPC (outside), with theoretical simulations (inside). The spatial character of the modes is clearly visile: Electron flow through the first mode shows one angular loe; flow through the second mode shows a V- shaped pattern with two angular loes and a zero down the center; and flow through the third mode shows three angular loes with two zeros. In addition, the experimental images show interference fringes, spaced y l F / (aout 0 nm), that demonstrate that the flow of electron waves is quantum mechanically coherent over the imaged distances. The spatial resolution of the images in figure 4 is excellent, considering the much larger size aout 100 nm across of the divot of depleted electron gas. How can such great resolution occur? The tip must ackscatter electrons arriving from the QPC in order to change the conductance and produce a signal. Imagine you are standing in a large room with lack walls, holding a flashlight against one side of your head with its eam pointed forward. To sense the pattern of flow of light from the flashlight, a friend holds a silver all in the light eam. Only a small glint of ackscattered light, much smaller than the all itself, will e visile to you. Similarly, the spatial resolution in images of electron flow is determined y the size of the glint, which is much smaller than the reflecting divot under the tip. At greater distances from the QPC, we discovered that the electron flow formed remarkaly narrow ranches, as shown in figure 5 for the first QPC conductance step. 16 From the experimental images and simulations close to the QPC shown in figure 4c, one might expect to see a single road angular loe of flow in figure 5 for the first mode of the QPC. Instead, the electron flow forms narrow ranches within distances from the QPC of much less than the electron mean free path, which is 11 mm for this sample. The ranches in images of electron flow are reproducile as long as the sample is kept at liquid-he temperatures. Fringes spaced y l F / are oserved over the entire image; their presence again underscores the coherent wavelike nature of electron transport in DEGs over large distances at low temperatures. The formation of ranches is a generic feature of electron flow in a DEG and is associated with small-angle scattering of electrons y the random potential induced y ionized donor atoms in the donor layer (see figure 1). In macroscopic measurements of the conductance, the electron diffusion constant in DEGs at low temperatures is determined y small-angle scattering. As shown in figure 1, the scattered electrons travel along smoothly curved paths rather than straight lines. Because the positions of the ionized donors remain fixed at low temperatures during the time required to record an image, these curved paths can e visualized. Thus one can image spatial structures, like the ranches in figure 5, that would not e visile in an ensemle average over many samples. Decemer 003 Physics Today 51

6 a Figure 7. Erasale electrostatic lithography can faricate customized two-dimensional electron gas devices. (a) A scanning proe microscope tip held at a negative voltage has deposited dots of charge (red) on the surface aove a DEG. The deposited charges, along with faricated electrodes (gray), define the walls of the DEG structure (yellow). () As the charged SPM tip is scanned aove the area outlined in green in panel (a), the conductance of the DEG shows a dip when the tip is over the quantum point contact defined y the deposited charge. (Adapted from ref. 18.) Ray-tracing simulations of the classical trajectories for electrons in an accurate depiction of the potential landscape in the DEG gives results like figure 6c, which shows a ranching structure similar to the experiments. 16 Branches are also clearly seen in the this month s cover image, which shows the simulated paths of electron trajectories. Closer inspection of the ray-tracing simulations reveals a lensing effect that, in hindsight, might have een expected. The hills and dales in the effective potential through which the electrons in the DEG pass deflect the electrons this way and that, occasionally inducing V- shaped cusps in the flow that correspond to folds in the position and momentum phase space for electrons. The simulation in figure 6a shows an example of a cusp produced y a single ionized donor atom. For scattering y many ionized donor atoms, cusps form at different locations, as shown in the simulation in figure 6. V-shaped cusps also appear in the experimental data; the arrow in figure 5 highlights an example. Once formed, cusps give rise to accumulations of trajectories that are hard to disperse, and those accumulations that are lucky (depending on the staility of susequent motion) carry quite a it of the flux and wind up looking like ranches. This microscopic ranching structure is perfectly consistent with the macroscopically measured mean free path and an exponential decay of momentum correlation in the diffusion of electron waves. An imaging revolution New ways to form and image quantum structures are developing rapidly. For example, Smith and Ritchie recently used an SPM tip to produce quantum structures in a DEG with erasale electrostatic lithography. 18 In that technique, an SPM tip is used to deposit charge on the surface of a heterostructure at low temperatures. Negative charge tends to deplete the underlying DEG and oppose the flow of electrons, and thus it creates the walls for a quantum structure. Figure 7 shows the location of the deposited charge defining a QPC; a conductance dip oserved when the charged SPM tip is scanned aove the QPC confirms the QPC s location. Erasale lithography provides a new G ( e / h) 1 1 mm 1 mm aility to change the geometry of a quantum structure during an experiment. The success that a numer of groups have had in imaging the flow of electrons the lightest y far of all the easily accessile particles through a DEG involves a sutle change of mindset, away from the ensemle and toward the individual. In condensed matter physics, there is a strong tradition of considering ensemle averages, ut now researchers are increasingly confronted with a specific quantum structure, with all its warts and umps. For example, the ranching of electron flow in DEGs is entirely consistent with earlier work using macroscopic measurements of conductance. But the newly discovered ranches put a face on the real agent of momentum decorrelation and even tell where donor atoms may e located in a particular sample. References 1. T. Chakraorty, P. Pietilainen, eds., The Quantum Hall Effects: Integral and Fractional, nd ed., Springer-Verlag, New York (1995).. L. L. Sohn, L. P. Kouwenhoven, G. 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