Scanning tunneling spectroscopy (STS), an extension of scanning tunneling microscopy (STM), provides information about the density of electrons in a sample as a function of their energy. In scanning tunneling microscopy, a metal tip is moved over a conducting sample without making physical contact. A bias voltage applied between the sample and the tip allows current to flow between them. This is a result of quantum tunneling across a barrier; in this instance, the physical distance between the tip and the sample. The scanning tunneling microscope is used to obtain "topographs" - topographic maps - of surfaces. The tip is rastered across a surface and, in constant current mode, a constant current is maintained between the tip and the sample by adjusting the height of the tip. A plot of the tip height at all measurement positions provides the topograph. These topographic images can obtain atomically resolved information on metallic and semiconducting surfaces. However, the scanning tunneling microscope does not measure the physical height of surface features. One such example of this limitation is an atom adsorbed onto a surface. The image will result in some perturbation of the height at this point. A detailed analysis of the way in which an image is formed shows that the transmission of the electric current between the tip and the sample depends on two factors: (1) the geometry of the sample and (2) the arrangement of the electrons in the sample. The latter is described quantum mechanically by an "electron density". The electron density is a function of both position and energy, and is formally described as the local density of electron states, abbreviated as local density of states (LDOS), which is a function of energy. Spectroscopy, in its most general sense, refers to a measurement of the number of something as a function of energy. For scanning tunneling spectroscopy the scanning tunneling microscope is used to measure the number of electrons (the LDOS) as a function of the electron energy. The electron energy is set by the electrical potential difference (voltage) between the sample and the tip. The location is set by the position of the tip. At its simplest, a "scanning tunneling spectrum" is obtained by placing a scanning tunneling microscope tip above a particular place on the sample. With the height of the tip fixed, the electron tunneling current is then measured as a function of electron energy by varying the voltage between the tip and the sample (the tip to sample voltage sets the electron energy). The change of the current with the energy of the electrons is the simplest spectrum that can be obtained, it is often referred to as an I-V curve. As is shown below, it is the slope of the I-V curve at each voltage (often called the dI/dV-curve) which is more fundamental because dI/dV corresponds to the electron density of states at the local position of the tip, the LDOS.
Introduction
Scanning tunneling spectroscopy is an experimental technique which uses a scanning tunneling microscope (STM) to probe the local density of electronic states (LDOS) and the band gap of surfaces and materials on surfaces at the atomic scale. Generally, STS involves observation of changes in constant-current topographs with tip-sample bias, local measurement of the tunneling current versus tip-sample bias (I-V) curve, measurement of the tunneling conductance, d I / d V {\displaystyle dI/dV} , or more than one of these. Since the tunneling current in a scanning tunneling microscope only flows in a region with diameter ~5 Å, STS is unusual in comparison with other surface spectroscopy techniques, which average over a larger surface region. The origins of STS are found in some of the earliest STM work of Gerd Binnig and Heinrich Rohrer, in which they observed changes in the appearance of some atoms in the (7 x 7) unit cell of the Si(111) – (7 x 7) surface with tip-sample bias. STS provides the possibility for probing the local electronic structure of metals, semiconductors, and thin insulators on a scale unobtainable with other spectroscopic methods. Additionally, topographic and spectroscopic data can be recorded simultaneously.
Tunneling current Since STS relies on tunneling phenomena and measurement of the tunneling current or its derivative, understanding the expressions for the tunneling current is very important. Using the modified Bardeen transfer Hamiltonian method, which treats tunneling as a perturbation, the tunneling current ( I {\displaystyle I} ) is found to be
where f ( E ) {\displaystyle f\left(E\right)} is the Fermi distribution function, ρ s {\displaystyle \rho _{s}} and ρ T {\displaystyle \rho _{T}} are the density of states (DOS) in the sample and tip, respectively, and M μ ν {\displaystyle M_{\mu \nu }} is the tunneling matrix element between the modified wavefunctions of the tip and the sample surface. The tunneling matrix element,
describes the energy lowering due to the interaction between the two states. Here ψ {\displaystyle \psi } and χ {\displaystyle \chi } are the sample wavefunction modified by the tip potential, and the tip wavefunction modified by sample potential, respectively. For low temperatures and a constant tunneling matrix element, the tunneling current reduces to
which is a convolution of the DOS of the tip and the sample. Generally, STS experiments attempt to probe the sample DOS, but equation (3) shows that the tip DOS must be known for the measurement to have meaning. Equation (3) implies that
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