Imaging Electron Flow
havior to be 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 difMark A. Topinka, Robert M. Westervelt, and Eric J. Heller
ficulty of designing and fabricating an
integrated circuit from a silicon crysemiconductor heterostructures have revolutionized tal without the use of an optical or electron microscope.
solid-state physics and its applications. Most of us use As device sizes continue to decrease, quantum behavior bethe fruits of this revolution every day in CD and DVD comes important and offers new research and application
recorders and players, cellular telephones, laser-based opportunities. To understand the fundamental behavior of
telecommunications, satellite television, and much more. electrons in this quantum regime and to make functioning
The technology, based on atomic layer-by-layer growth devices based on this behavior, one must develop ways to
using molecular beam epitaxy (MBE), is sophisticated, re- visualize the flow of electron charges and spins through
markable, and marketable.
semiconductors. The invention of the scanning tunneling
One class of semiconductor heterostructures, the two- microscope (STM) allowed researchers to directly view
dimensional electron gas (2DEG), has been a focal point the pattern of atoms on a material’s surface. Additional
for theorists and experimentalists, and a wellspring of new methods are needed to image the flow of electrons beneath
physics. A 2DEG can be produced at low temperatures at the surface.
an interface of two distinct layers (a so-called heterojuncObtaining images of 2DEGs inside semiconductors is
tion) doped nearby with atoms that donate electrons. The no easy matter, because the electrons are buried beneath
electrons at such a junction are confined to the lowest the surface and because the samples must be cooled to low
quantum state in the direction normal to the interface; by temperatures to show quantum behavior. Nonetheless, a
charging gate electrodes on the top surface of the het- number of groups have recently developed liquiderostructure some distance away to repel them, the elec- helium–cooled scanning probe microscopes (SPM) for this
trons can be further confined in the other directions to purpose.
make dots, wires, resonators, and other shapes. That technology has led to celebrated discoveries including the in- Making a two-dimensional electron gas
teger and fractional quantum Hall effect (QHE),1 the Figure 1 illustrates how a 2DEG can be created inside a
Coulomb blockade and single-electron transistors, and gallium arsenide–aluminum gallium arsenide hetconductance quantization in quantum point contacts erostructure. During growth by MBE, atomic layers are
(QPCs).2 The potential for exploiting these and many other added to the heterostructure at a rate of about one layer
quantum effects is spawning new fields of single electron- per second. Conduction-band profiles can be engineered
ics3 and spintronics4—new approaches to logic that use sin- during growth by changing the Al fraction. Switching
gle electron charges and spins to represent bits of data— abruptly from GaAs to AlGaAs creates a very clean, sharp
and the new area of quantum information processing, barrier that keeps electrons inside the GaAs layer. Silicon
based on the coherent interaction of quantum mechanical atoms in the AlGaAs layer act as donors: The electrons ionqubits.5
ized from the Si fall over the barrier and are trapped at
Despite all the beautiful experiments already per- the GaAs–AlGaAs interface, where they can move freely
formed on 2DEGs and all that is riding on the new science for long distances. At low temperatures, movement in the
and phenomena made possible by them, researchers have z-direction (normal to the interface) is frozen out, but the
been blind until recently as to how electrons actually move electrons, trapped in a 2D “flatland” sheet, are free to move
through them. Most of the knowledge of electron flow in and interact in the x- and y-directions.
2DEGs is indirect, based on electron-transport measureTwo-dimensional electron gases possess a unique comments of macroscopically averaged quantities. To be sure, bination of parameters that together make an ideal labomany of the statistical properties are known, such as the ratory in which to explore the fascinating and often surelectron mean free path. But macroscopically averaged pa- prising behavior of such systems. The electron’s de Broglie
rameters do not reveal the details of the fascinating be- wavelength at the Fermi energy is unusually long, typically 20–100 nm, and it can be an appreciable fraction of
Mark Topinka, now a Urbanek Postdoctoral Fellow in the applied
the size of a typical device. (Electron wavelengths in metphysics department at Stanford University, received his PhD from
als, by comparison, are typically well under 1 nm.) The
Harvard University under Bob Westervelt, Mallinckrodt ProfesFermi wavelength can be tuned by changing the electron
sor of Applied Physics and of Physics. Rick Heller is a professor
density with a planar gate electrode located beneath the
of chemistry and physics at Harvard.
2DEG. Because they do not collide often with other parti-
New scanning probe techniques provide fascinating
glimpses into the detailed behavior of semiconductor
devices in the quantum regime.
S
© 2003 American Institute of Physics, S-0031-9228-0312-020-0
December 2003
Physics Today
47
Figure 1. A two-dimensional
electron gas formed at the interface
between gallium arsenide and
aluminum gallium arsenide in a
semiconductor heterostructure. The
AlGaAs layer (green) contains a
layer (purple) of silicon donor atoms
(dark blue). Electrons from the donor
layer fall into the GaAs layer (pink)
to form a 2DEG (blue) at the interface. The ionized Si donors (red)
create a potential landscape for the
electron gas; the resulting smallangle scattering smoothly bends
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
2DEG for tens or even hundreds of microns before losing track of their initial direction.
The motion of electrons in a GaAs–AlGaAs
2DEG is limited by small-angle scattering. Positively
charged donor ions create a bumpy 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 but randomly varying index
of refraction, electrons travel through the 2DEG along
smoothly bent paths. Small-angle scattering is an elastic
process, because 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 buffeted 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 2DEG.1 In a strong magnetic field applied perpendicular to the 2DEG, the electrons no longer travel as
free plane waves but instead occupy a series of discrete
Landau levels separated in energy by \wc, the energy associated with the cyclotron frequency wc (\ is Planck’s constant). Both the number of states in each Landau level and
the spacing between the levels scale with the applied field.
Motion along the field lines cannot occur, because 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 observed. Quantum
Hall plateaus occur at magnetic fields for which the highest occupied Landau level is almost completely filled. The
number of filled Landau levels is called the filling factor n.
On a quantum Hall plateau, electrons in the middle
of the sample form an incompressible liquid (see PHYSICS
TODAY, August 2003, page 38). The longitudinal resistance
of the sample goes to zero, and the transverse Hall resist48
December 2003
Physics Today
ance is quantized on
plateaus of height (1/n)h/e2
with integer values of the filling factor
n, where e is the electron charge. All of the current through the sample is carried by edge states
that pass around its circumference. These edge states
correspond to classical skipping orbits 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/e2 for certain
fractional numbers, such as 1/3, is also associated with
edge states, but its source is the correlated motion of electrons in the 2DEG.
The QHE is associated with remarkable spatial structures in the 2DEG. The structures include predicted spatially striped phases of the quantum Hall liquid that have
been investigated using macroscopic measurements. Imaging the properties of a 2DEG in a strong magnetic field at
low temperatures is a particularly useful way to understand
the QHE, and a number of groups have developed cooled
scanning probe 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 subsurface charge accumulation (SCA) probe.6 An STM
tip held above the surface capacitively couples to the 2DEG
immediately below. When a small AC voltage is applied between the tip and the 2DEG, the resulting flow of charge
in the gas induces an oscillating image charge on the tip;
that oscillation is detected by a sensitive electrometer.
Adding a positive DC voltage to the tip allows spatial profiling of the 2DEG by creating a small bubble composed of
a few electron charges beneath the tip. The bubble is surrounded by an insulating ring of incompressible fluid in
the quantum Hall regime, and it forms an electrically isolated quantum dot that holds a fixed, discrete number of
electron charges.7
Figure 2 is an image of the SCA signal obtained as the
bubble is moved through the 2DEG by scanning the STM
tip above the surface. As the number of electrons in the
bubble changes, the signal intensity oscillates, which results in bright strips that form a contour map of the random electrostatic potential inside the quantum Hall liquid.
http://www.physicstoday.org
Figure 2. Subsurface charge accumulation in a two-dimensional electron gas maps the electrostatic potential experienced by the 2DEG in the quantum Hall regime with a filling factor n ⊂ 1. A positive voltage on the tip of a scanning
tunneling microscope above the sample pulls in electrons to
create a few-electron bubble in the 2DEG. The closed contours in this 2.5 × 2.5-mm2 image are caused by the quantization of electronic charge inside the bubble: The contours,
which arise as individual electrons move in and out of the
bubble, surround high and low regions of the random electrostatic potential. (Adapted from ref. 7.)
This image directly demonstrates how, thanks to the
2DEG’s incompressibility, the QHE can block the flow of
electrons inside the electron gas.
Amir Yacoby (now at the Weizmann Institute of Science) and his colleagues at Lucent Technologies’ Bell Labs
developed an alternate way to image the charge of an electron gas.8 They successfully fabricated a single-electron
transistor (SET) on the tip of a glass fiber and used it as
a scanning electrometer probe. The SET was sensitive
enough to detect changes in the 2DEG potential and density variations corresponding to tiny fractions of a single
electron. With their probe, 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 by measuring the charge induced on the SET elec-
trometer when a voltage was applied to a gate electrode
beneath the 2DEG. The incompressible strips of electron
gas next to the edge states allowed the electric field to pass
through the 2DEG to reach the SET; the result was the
bright strips in figure 3a. The SET tip was also used to
image the Hall potential, as shown in figure 3b for a magnetic field near the n = 2 plateau. The Hall potential appears at the edges of the 2DEG, but 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 Labs9 to image, in a manner similar to
the Ashoori group, localized 2DEG electron states in the
quantum Hall regime.
The quantum Hall regime imaging by the MIT and
Bell Labs teams—as well as by Paul McEuen10 at the University of California, Berkeley and Cornell University, by
Jürgen Weis and Klaus von Klitzing11 at the Max Planck
Institute for Solid State Research in Stuttgart, Germany,
and by Klaus Ensslin12 at ETH Zürich—represents
groundbreaking achievements that have enriched theories
of the QHE and revealed intriguing new effects.
Imaging electron flow in low magnetic fields
SURFACE POTENTIAL
CHANGE
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
by our group at Harvard University, by McEuen, by
+2.5 mV
a
b
Charles Smith and David
Ritchie at the University of
Cambridge, and by Ensslin
has focused on electron flow
patterns in a 2DEG 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
by directly imaging the mean
free path in a 2DEG for electrons passing through a wide
–2.5 mV
constriction.13 Four years
later, Rolf Crook and colleagues at Cambridge imFigure 3. Imaging with a single-electron transistor. (a) Edge states form along the
aged cyclotron orbits in a
boundaries of a two-dimensional electron gas in the quantum Hall regime. The bright
2DEG at 4.2 K and interstrips in this 15 × 15-mm2 image are incompressible regions of the electron gas next to
preted features of their imthe edge state for a filling factor n ⊂ 2. The incompressible regions allow the electric
ages in terms of the deflecfield from an electrode beneath the 2DEG to reach the SET. (Negative voltage applied to
tions in electron trajectories
the two black gate electrodes has closed the constriction between them.) (b) The SET
caused by donor atom density
probe can also detect changes in potential. A current flowing through an open aperture
fluctuations and impuribetween the electrodes generates a transverse Hall potential that appears along the
ties.14 Using a charged
edges of the electrodes. No longitudinal potential is apparent at the top and bottom of
atomic-force microscope tip
this 7 × 7-mm2 image. (Adapted from ref. 8.)
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December 2003
Physics Today
49
c
a
200 nm
d
CONDUCTANCE (e2/h)
b
6
200 nm
4
e
2
0
–1.2 –1.1
–1.0 –0.9
GATE VOLTAGE (mV)
200 nm
to bend the trajectories of electrons traveling between two
QPCs, they achieved glimpses into the spatial details of
electron flow in 2DEG nanostructures. Those early experiments set the scene for more recent high-resolution images that revealed surprising and important details about
electron flow in 2DEGs.
At Harvard, we used scanning probe microscopy to
image the coherent flow of electron waves through a 2DEG
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 QPC17, a narrow constriction whose width is comparable to the electrons’ Fermi
wavelength lF (see PHYSICS TODAY, July 1996, page 22). As
its width is increased, the conductance of a QPC increases
in steps of height 2e2/h because the electrons travel
through individual transverse modes, each of which con-
December 2003
tributes 2e2/h to the total conductance.
Figure 4a illustrates our technique for imaging the
flow of electron waves in a GaAs–GaAlAs 2DEG. Gates on
the surface form a QPC whose width could be adjusted by
changing the gate voltage. A charged SPM tip capacitively
couples to the electron gas below; for negative tip-to-gas
voltages, it can deplete a small, round divot in the 2DEG
that reflects electron waves arriving from the QPC. The
pattern of electron waves scattered by the divot under the
tip is shown by theoretical simulations in the figure. Some
of the electrons reflected by the divot return along their incoming path and travel back through the QPC, measurably reducing its conductance. Electrons scattered at other
angles have little effect on the conductance because they
remain on the same side of the QPC.
The change in conductance induced by
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
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 branches that are
produced by 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 by the
focusing effect of a nearby impurity
atom. (Adapted from ref. 16.)
1 mm
50
Figure 4. Electron flow through a quantum point contact. (a) Scheme for imaging current flow through a QPC using
scanning probe microscopy. Two gate
electrodes (yellow) create a narrow constriction in the underlying two-dimensional electron gas. A charged tip (green)
depletes the electron gas below it, creating a divot (red spot) that scatters incoming electron waves, as shown in the simulations (blue). (b) 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
below each step show simulations of the
spatial pattern of electron flow for the
transverse modes that contribute 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 observed interference fringes
spaced by half the Fermi wavelength
demonstrate the coherence of electron
flow. Because the additional flow, appearing as the QPC becomes wider, is
due to the newly opened-up mode, the
image for each transverse mode could be
obtained by subtracting the raw images from the next
lower step. (Adapted from ref. 15.)
Physics Today
http://www.physicstoday.org
a
b
c
Figure 6. Simulations of electron flow. (a) Parallel electron trajectories, going from left to right, form a V-shaped cusp due to
focusing by a potential-energy dip caused by a charged donor atom (not seen) above a two-dimensional electron gas. (b) A
realistic 2DEG 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 2DEG quantum point contact samples. The potential is
shown green in the valleys and white on the peaks. The electron flux is coded by height and color, with blue 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 branches. (S. E. J. Shaw, PhD thesis, Harvard University, 2002.)
electron flux that was there before 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 responsible for the conductance steps
shown in figure 4b.15 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 visible: Electron flow through the first mode shows one angular lobe; flow through the second mode shows a Vshaped pattern with two angular lobes and a zero down
the center; and flow through the third mode shows three
angular lobes with two zeros. In addition, the experimental images show interference fringes, spaced by lF /2
(about 20 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—about 100 nm
across—of the divot of depleted electron gas. How can such
great resolution occur? The tip must backscatter electrons
arriving from the QPC in order to change the conductance
and produce a signal. Imagine you are standing in a large
room with black walls, holding a flashlight against one side
of your head with its beam pointed forward. To sense the
pattern of flow of light from the flashlight, a friend holds a
silver ball in the light beam. Only a small glint of backscattered light, much smaller than the ball itself, will be visible to you. Similarly, the spatial resolution in images of
electron flow is determined by the size of the glint, which
http://www.physicstoday.org
is much smaller than the reflecting divot under the tip.
At greater distances from the QPC, we discovered that
the electron flow formed remarkably narrow branches, 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 broad angular lobe of flow in figure 5 for the first mode
of the QPC. Instead, the electron flow forms narrow
branches within distances from the QPC of much less than
the electron mean free path, which is 11 mm for this sample. The branches in images of electron flow are reproducible as long as the sample is kept at liquid-He temperatures. Fringes spaced by lF /2 are observed over the entire
image; their presence again underscores the coherent
wavelike nature of electron transport in 2DEGs over large
distances at low temperatures.
The formation of branches is a generic feature of electron flow in a 2DEG and is associated with small-angle
scattering of electrons by the random potential induced by
ionized donor atoms in the donor layer (see figure 1). In
macroscopic measurements of the conductance, the electron diffusion constant in 2DEGs at low temperatures is
determined by 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 be visualized. Thus one can image spatial structures, like the branches in figure 5, that would not be visible in an ensemble average over many samples.
December 2003
Physics Today
51
2
b
G (2e2/h)
a
1 mm
1
Figure 7. Erasable electrostatic lithography can fabricate
customized two-dimensional electron gas devices. (a) A
scanning probe microscope tip held at a negative voltage
has deposited dots of charge (red) on the surface above a
2DEG. The deposited charges, along with fabricated electrodes (gray), define the walls of the 2DEG structure (yellow). (b) As the charged SPM tip is scanned above the area
outlined in green in panel (a), the conductance of the 2DEG
shows a dip when the tip is over the quantum point contact
defined by 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 2DEG gives results like figure 6c, which shows a
branching structure similar to the experiments.16 Branches
are also clearly seen in the this month’s cover image, which
shows the simulated paths of 50 000 electron trajectories.
Closer inspection of the ray-tracing simulations reveals a lensing effect that, in hindsight, might have been
expected. The hills and dales in the effective potential
through which the electrons in the 2DEG pass deflect the
electrons this way and that, occasionally inducing Vshaped 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 by a single ionized donor atom. For scattering by
many ionized donor atoms, cusps form at different locations, as shown in the simulation in figure 6b. 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 stability of subsequent motion) carry quite a bit
of the flux and wind up looking like branches. This microscopic branching 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 2DEG
with erasable 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 2DEG 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 observed when
the charged SPM tip is scanned above the QPC confirms
the QPC’s location. Erasable lithography provides a new
52
December 2003
Physics Today
ability to change the geometry of a quantum structure
during an experiment.
The success that a number of groups have had in imaging the flow of electrons—
the lightest by far of all the
easily accessible particles—
through a 2DEG involves a
subtle change of mindset,
away from the ensemble and
toward the individual. In
1 mm
condensed matter physics,
there is a strong tradition of
considering ensemble averages, but now researchers
are increasingly confronted with a specific quantum structure, with all its warts and bumps. For example, the
branching of electron flow in 2DEGs is entirely consistent
with earlier work using macroscopic measurements of conductance. But the newly discovered branches put a face on
the real agent of momentum decorrelation and even tell
where donor atoms may be located in a particular sample.
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