Transcript Theory & Implementation of the Scanning Tunneling Microscope

Theory & Implementation of the
Scanning Tunneling Microscope
Neil Troy
The Scanning Tunneling Microscope (STM)
• Classical vs. Quantum mechanics
• Engineering Hurdles
• Implementation & Images
STM invented in 1981 by Greg Binnig & Heinrich Rohrer at IBM and were
awarded with a Nobel Prize in 1986
Classical vs. Quantum
Classically, if one threw a ball at a brick wall (and didn’t miss horribly) they would
always find the ball on their side of the wall, because unless you are Superman there is
no way that you are throwing the ball hard enough to break through the wall.
Some
later
time
Quantum mechanically, however, the ball has some finite probability of “tunneling”
through the wall. The ball need not be thrown incredibly hard to achieve this,
although the more energy the ball has the more likely it will be to tunnel.
Some
later
time
Mathematical Approach
Standard 1-D wave equation where U(z) is the potential of the barrier, and
E is the particle’s energy
 2  2 n ( z )

 U ( z ) n ( z )  E n ( z )
2
2m z
Which has the normal solution of a traveling wave:
 n ( z )   n (0)e  ikz
where
k
2m( E  U )
2
This is the general solution but in the event that U > E we can factor out an
i and we get a solution that has a completely different meaning.
 n ( z )   n ( 0) e
 kz
which for our scenario, where z > 0, we get an exponential decay as the
electron sees if it can tunnel through the barrier.
Visualization of tunneling
Lowenergy
energyparticle
particlehas
is confined
to of
Higher
a possibility
stay the
in its
potential
well.
passing through
wall
and continuing
with less
energy than before.
Practical application for probing
A
Current
0
If two metals are connected to opposite ends of a battery but are
separated from each other than they logically should not conduct
electricity. However, as we bring the metals extremely close to each other
the wavefunctions of the electrons are able to tunnel through the gap and
we will detect a current on our meter.
How this is different than arcing
Let’s abandon
mechanics
and our
microscope
for a second
Thisquantum
is drastically
different
than
tunneling because
the and look
at thehigh
slightly
different
concept of
arc/lightning.
extremely
voltages
are physically
changing
the transfer
medium (ionizing) so that the electrons can conduct. As such
the arcing could be continuous (if the materials weren’t
Only
at a certain
distance
thecurrent
voltagewould
goingchange
to arc through
normally
destroyed)
butisthe
since thethe
media (air). This could
be repeated
and changing.
as long as the environment
medium
is constantly
hasn’t changed it is very repeatable, but not practical.
Hi Voltage
Problems with probabilities
After a little bit of math we can come up with the probability that an
electron can tunnel through our barrier,
P   n (0) e  2 kW
2
where W is the barrier width.
We can look at the above probability as the chance of an electron on one
metal being found on the other, but likewise this works the opposite way as
well. This brings rise to another problem with the above equation, even if an
electron could tunnel the gap it has to have a home on the other metal.
First, let’s look at biasing one movement over the gap. This is achieved just
as shown previously by connecting both metals to different ends of a
voltage source. This voltage is very important though, we don’t want to arc
the material (high voltage) but we do want to give the electrons a natural
tendency to flow one way. As such, the applied voltage is normally on the
order of the work function of the material (a few eV, 4-5eV).
So many electrons to count
Since we are applying a voltage nearly equivalent to the work function of
the material, we are allowing any electron at, or near, the fermi energy the
possibility of being free and tunneling our gap.
We are now dealing with fermi energies and as such our talk must now
change from individual electrons but to the masses. We now have to look at
all the possible electrons that we could see tunneling our gap. Luckily, we
have some bounds on these electrons, so we need not look at all of them.
Ef
Ef-eV
To be more exact we care only about a
small band of electrons that are very
close to the fermi energy of the material.
If we were to measure the probability as
the metals are brought close we could only
possibly detect those electrons move, and
this is only a half truth since we still
require they have a free state to tunnel
into.
Currents, a more usable quantity
Unfortunately, my probability meter is broken so we need some other
measure of electrons moving across our gap. Electrons moving leads
instantly into current so we need a way to quantify our probabilities in
terms of currents.
Without getting too complicated or mathematical I give you a relatively
simple equation of the current we can expect to detect:
I  V (W , E f )e2kW
where (W,Ef) is the density of states of one metal through the gap,
W, at the other metal.
Some engineering concerns
With some physics in the bag let’s look at some engineering challenges. We
need an extremely accurate way to bring a probing material to our sample,
that is stable and is capable of very fine adjustments. To give you a feel for
if one cranked out some numbers for the above equations we need to be
able to bring our probe to within Angstroms of the surface and then be
able to move in Angstrom increments.
Sample
Angstroms
Probe
Some engineering concerns (cont.)
A second problem is that we cannot simply move a block of metal towards a
sample, if we did we would be probing the entire surface at once and that is
only if our surface is perfectly flat!
Sample
Probe
We’ll look at this one first.
A perfect probe
Hopefully, it is clear that we should have a probe that is extremely small, or
ideally on the atomic scale. If we were to look at the math again (which I’m
not) we would see that if we had a probe that had exactly one atom on the
end then the subsequent atoms behind it would contribute almost nothing to
our current.
Current
~e-2(6)k
~2e-2(9)k
6Å
Sample
3Å
As it turns out atomic probes are easily
made with either mechanical or chemical
processes.
Accurate distance control
Angstrom level distance control is required which would be impossible for
any mechanical gear workings.
Enter the world of piezoelectrics
A piezoelectric is a crystal that creates potential differences (voltages)
when mechanical stresses are imposed on it.
Voltage
0
Piezoelectric motors
The mechanical to electrical effect is completely reversible and by applying
voltages to these materials we force mechanical effects out of them.
In fact this effect is so common place that we can find it in everyday
objects like speakers to printers, and they can be as sophisticated as
extremely accurate mirror control in laser laboratories.
One can reliably place our sampling tip within 10’s of Angstroms from a
surface without using piezoelectric motors but beyond that the accurate
probing is handled by the piezoelectric.
We’ll even use these motors in a feedback system to adjust the height of
our probe.
Probing a surface
With all of the elements in place we can start probing our metallic surface.
Height
Current
Or
Wewe
could
could
either
vary probe
the height
at a constant
in search
height and
for look
constant
at current
current.
variations
as we pass.
PROBE
Distance
Probing a surface & what it says
Although more complex, one normally modulates the height of the probe.
We can see this as a major advantage in the sense that we are not
contingent on the surface being very uniform, a crag or cliff of atoms does
not pose a collision hazard as well as at a constant current we can suppress
effects that may come with higher currents (heat, breakdown of tip, etc.).
The biggest thing we must keep in mind is that an STM cannot “see” atoms.
An STM probes the density of states of a surface.
As such, holes are a common artifact of STM, a hole does not mean that no
atom exists it merely means that the probe cannot reach below the first
layer of atoms. Likewise, peaks in an STM do not necessarily mean there is
a mountain of atoms, the atoms may be a different element from the
surroundings and may possess a higher density of states resulting in a
higher current/probe height.
Images of an STM
The Nobel prize winning first STM.
A more current example.
STM produced images
STM image, 7 nm x 7 nm, of a
single zig-zag chain of Cs atoms
(red) on the GaAs(110) surface
(blue). *
*National Institute of Standards and Technology
Surface of platinum.
IBM, Almaden Research Facility
Surface of nickel.
IBM, Almaden Research Facility
Surface of copper.
IBM, Almaden Research Facility
Ring of iron atoms on a copper blanket.
IBM, Almaden Research Facility
Startling aside...
In 1989, at the Almaden IBM Research Facility scientists found
that an STM could be used to lift atoms off a surface of metal and
placed back in a different location, at low temperatures that is.