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.K. Oura, V. G. Lifshits, A. A. Saranin, A. V. Zotov, and M. Katayama, Surface Science: An Introduction, Berlin: Springer-Verlag, 2003. 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, , 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.R. J. Hamers and D. F. Padowitz, “Methods of Tunneling Spectroscopy with the STM,” from Scanning Probe Microscopy and Spectroscopy: Theory, Techniques, and Applications, 2nd ed., Ed. by D. A. Bonnell, New York: Wiley-VCH, Inc., 2001. 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 () is found to be
where is the Fermi distribution function, and are the density of states (DOS) in the sample and tip, respectively, and 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 and are the sample wavefunction modified by the tip potential, and the tip wavefunction modified by sample potential, respectively.C. Julian Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press New York (1993).
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
under the gross assumption that the tip DOS is constant. For these ideal assumptions, the tunneling conductance is directly proportional to the sample DOS.
For higher bias voltages, the predictions of simple planar tunneling models using the Wentzel-Kramers Brillouin (WKB) approximation are useful. In the WKB theory, the tunneling current is predicted to be
where and are the density of states (DOS) in the sample and tip, respectively. The energy- and bias-dependent electron tunneling transition probability, T, is given by
where and are the respective work functions of the sample and tip and is the distance from the sample to the tip.
The tip is often regarded to be a single molecule, essentially neglecting further shapes induced effects. This approximation is the Tersoff-Hamann approximation, which suggests the tip to be a single ball-shaped molecule of certain radius. The tunneling current therefore becomes proportional to the local density of states (LDOS).
Experimental methods Acquiring standard STM topographs at many different tip-sample biases and comparing to experimental topographic information is perhaps the most straightforward spectroscopic method. The tip-sample bias can also be changed on a line-by-line basis during a single scan. This method creates two interleaved images at different biases. Since only the states between the Fermi levels of the sample and the tip contribute to , this method is a quick way to determine whether there are any interesting bias-dependent features on the surface. However, only limited information about the electronic structure can be extracted by this method, since the constant topographs depend on the tip and sample DOS's and the tunneling transmission probability, which depends on the tip-sample spacing, as described in equation (5).R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy: Methods and Applications, Cambridge, UK: Cambridge University Press, 1994.
By using modulation techniques, a constant current topograph and the spatially resolved can be acquired simultaneously. A small, high frequency sinusoidal modulation voltage is superimposed on the D.C. tip-sample bias. The A.C. component of the tunneling current is recorded using a lock-in amplifier, and the component in-phase with the tip-sample bias modulation gives directly. The amplitude of the modulation Vm has to be kept smaller than the spacing of the characteristic spectral features. The broadening caused by the modulation amplitude is 2 eVm and it has to be added to the thermal broadening of 3.2 kBT. In practice, the modulation frequency is chosen slightly higher than the bandwidth of the STM feedback system. This choice prevents the feedback control from compensating for the modulation by changing the tip-sample spacing and minimizes the displacement current 90° out-of-phase with the applied bias modulation. Such effects arise from the capacitance between the tip and the sample, which grows as the modulation frequency increases.
In order to obtain I-V curves simultaneously with a topograph, a sample-and-hold circuit is used in the feedback loop for the z piezo signal. The sample-and-hold circuit freezes the voltage applied to the z piezo, which freezes the tip-sample distance, at the desired location allowing I-V measurements without the feedback system responding.R. C. Barrett and S. Park, “Design Considerations for an STM System,” from Scanning Tunneling Microscopy, Ed. by W. J. Kaiser and J. A. Stroscio, San Diego: Academic Press, Inc., 1993. The tip-sample bias is swept between the specified values, and the tunneling current is recorded. After the spectra acquisition, the tip-sample bias is returned to the scanning value, and the scan resumes. Using this method, the local electronic structure of semiconductors in the band gap can be probed.
There are two ways to record I-V curves in the manner described above. In constant-spacing scanning tunneling spectroscopy (CS-STS), the tip stops scanning at the desired location to obtain an I-V curve. The tip-sample spacing is adjusted to reach the desired initial current, which may be different from the initial current setpoint, at a specified tip-sample bias. A sample-and-hold amplifier freezes the z piezo feedback signal, which holds the tip-sample spacing constant by preventing the feedback system from changing the bias applied to the z piezo. The tip-sample bias is swept through the specified values, and the tunneling current is recorded. Either numerical differentiation of I(V) or lock-in detection as described above for modulation techniques can be used to find . If lock-in detection is used, then an A.C. modulation voltage is applied to the D.C. tip-sample bias during the bias sweep and the A.C. component of the current in-phase with the modulation voltage is recorded.
In variable-spacing scanning tunneling spectroscopy (VS-STS), the same steps occur as in CS-STS through turning off the feedback. As the tip-sample bias is swept through the specified values, the tip-sample spacing is decreased continuously as the magnitude of the bias is reduced. Generally, a minimum tip-sample spacing is specified to prevent the tip from crashing into the sample surface at the 0 V tip-sample bias. Lock-in detection and modulation techniques are used to find the conductivity, because the tunneling current is a function also of the varying tip-sample spacing. Numerical differentiation of I(V) with respect to V would include the contributions from the varying tip-sample spacing.R. M. Feenstra and J. A. Stroscio, “Methods of Tunneling Spectroscopy,” from Scanning Tunneling Microscopy, Ed. by W. J. Kaiser and J. A. Stroscio, San Diego: Academic Press, Inc., 1993. Introduced by Mårtensson and Feenstra to allow conductivity measurements over several orders of magnitude, VS-STS is useful for conductivity measurements on systems with large band gaps. Such measurements are necessary to properly define the band edges and examine the gap for states.