# STM Measurement Types

Our STM holds the tip at virtual ground, and applies a bias voltage
*V* to the sample. The tunneling current is measured from the tip.
The tip sits at the end of a piezo tube scanner (with 4 quadrants
±*x* and ±*y* on the outside, and a single electrode for
*z* on the inside), which can control its motion with sub-Angstrom
precision in 3 directions. So we can measure tunneling current as a
function of 4 variables: *x*, *y*, *z*, and *V*. In
practice, we usually attempt to keep *z* constant by employing a
feedback loop to keep the tunneling current constant at a fixed bias
voltage. Assuming that *z* is constant, we then take measurements as
a function of *x*, *y*, and *V*.

### Topography

The most common mode of STM measurement employed by STM groups around
the world is a topography. In this mode, we raster the tip across the
surface at a fixed sample bias voltage *V*_{set}, and employ
a feedback loop which controls the voltage on the *z* piezo to keep
the tunneling current constant at *I*_{set}. By recording the
voltage to the *z* piezo, we can effectively map the height of the
surface.

It's actually not clear what we mean by the "height" of the surface. One obvious suggestion would be some contour of constant charge density. However, as we can see from:

the tunneling current does not depend on the total
charge density, but only the charge density within *eV* below the
Fermi surface, where *-V* is the applied bias voltage.

One might argue then that we should apply an arbitrarily large voltage
so that we can capture more of the charge density, but we then run into
two problems: (1) Some of our samples are fragile compounds with weak
bonds, and if we apply a large voltage locally, pieces of the surface will
literally rip off. (2) If *V* is too high (on order the work function
φ), our tunneling approximation breaks down.

So we are stuck with this somewhat arbitrary definition of the "height
of the surface" as the tip-sample separation for which tunneling current
is fixed at a particular constant value *I*_{set} for a
particular applied bias voltage *V*set. In practice we
usually choose to fix the current at -100 pA, for a bias voltage of -100
mV. This is arbitrary, but gains some validity from the fact that we do
see atoms and other structural features as expected, even over a wider
range of choices of *I*_{set} and *V*_{set}. The
most widely varying density of states features of the superconducting
samples we have studied so far seem to be within 75 meV of the Fermi
level.

### Density of states

From the tunneling equation above, we see that if we hold the
tip-sample separation constant, at a given *(x, y)* location, and
put a negative bias voltage *-V* on the sample, we have:

In other words, we can measure the integral of the
density of states, down to any energy *-eV*, by varying *-V*.
Note that for a negative bias voltage on the sample, we are tunneling
electrons from sample to tip, and we are measuring the integrated density
of full states below the Fermi level in the sample. For a positive bias
voltage on the sample, we are tunneling electrons from tip to sample, and
we are measuring the integrated density of empty states above the Fermi
level in the sample.

OK, that's nice, we have the integrated density of states (IDOS).
But it would be much nicer to just have the density of states
(DOS). After acquiring an IDOS vs. *V* curve, we could take a
numerical derivative of our data to get the DOS. But taking a
derivative numerically is a horribly noisy thing to do. It is
much better to measure the derivative directly.

So we employ a lock-in amplifier to modulate the bias voltage by
*dV* (typically a few mV) around a DC voltage *V* of interest.
As a result of the voltage modulation *dV* we can measure a current
modulation *dI*. We call this *dI/dV* the conductance
*g(V)*. Now, we can write:

Therefore, by using a lock-in and varying *V*, we
can map out an entire density of states curve.

The energy resolution is limited by the amplitude of the wiggle (until
the modulation becomes less than approximately *K _{B}T* =
0.36 meV at

*T*= 4.2 K). So ideally, we could make the voltage modulation smaller than 0.36 mV. But in practice, we can't get enough signal-to-noise at this low amplitude without prohibitively long averaging times. Most of our data is measured with a 2 mV RMS modulation, therefore blurring our energy resolution by approximately 5.6 meV.

### Linecut

In the previous section, we discussed a single DOS curve at a
single location. Since we have *(x, y)* control over the location
of our tip using the piezo tube scanner, we can measure DOS curves
anywhere we want. Some samples, like good metals (without
impurities), should have a completely homogenous DOS everywhere.

But some more interesting samples are inhomogeneous. For example, we can measure a full DOS curve at every point along a straight line, spaced a few Å apart, and we see a "linecut".

### DOS map

Basically what we are discussing here is a three-dimensional data set:
two spatial dimensions *x* and *y* (by varying the position of
the tip) and one energy dimension (by varying *V*). We can view this
3-dimensional data set as a series of DOS-vs-energy curves at every
location *(x, y)*, or we could view it as a series of 2-dim DOS-maps
at each energy *eV*. Mapping the DOS at a specific energy is a good
visual way to see the inhomogeneities in the density of states.