Scanning tunneling spectroscopy

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, 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) is found to be {{NumBlk|| I = \frac{4\pi e}{\hbar}\int_{-\infty}^{\infty}\left[f\left(E_F-eV+\varepsilon\right)-f\left(E_F+\varepsilon\right)\right] \rho_S\left(E_F-eV+\varepsilon\right) \rho_T\left(E_F+\varepsilon\right) \left|M_{\mu\nu}\right|^2 \, d\varepsilon \ ,|}} where f\left(E\right) is the Fermi distribution function, \rho_s and \rho_T are the density of states (DOS) in the sample and tip, respectively, and M_{\mu\nu} is the tunneling matrix element between the modified wavefunctions of the tip and the sample surface. The tunneling matrix element, {{NumBlk|| M_{\mu\nu} = -\frac{\hbar^2}{2m} \int_\Sigma\left( \chi_\nu^*\nabla\psi_\mu - \psi_\mu \nabla \chi_\nu^*\right) \cdot \,d{\mathbf {S}}\ ,|}} describes the energy lowering due to the interaction between the two states. Here \psi and \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 {{NumBlk|| I \propto\int_0^{eV} \rho_S\left(E_F-eV+\varepsilon\right) \rho_T\left(E_F+\varepsilon\right) \, d\varepsilon\ ,|}} 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 {{NumBlk|| \frac{dI}{dV} \propto\rho_S\left(E_F-eV\right)\ ,|}} 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 {{NumBlk|| I \propto \int_0^{eV}\rho_S\left(r,E\right)\rho_T\left(r,E-eV\right) T\left(E,eV,r\right)\,dE\ ,|}} where \rho_s and \rho_t 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 {{NumBlk||T=\exp\left(- \frac{2Z\sqrt{2m}}{\hbar}\sqrt{\frac{\phi_s+\phi_t}{2}+\frac{eV}{2}-E}\right)\ ,|}} where \phi_s and \phi_t are the respective work functions of the sample and tip and Z 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 I, 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 I 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). By using modulation techniques, a constant current topograph and the spatially resolved dI/dV 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 dI/dV 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. 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. 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. 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. Because the topographic image and the tunneling spectroscopy data are obtained nearly simultaneously, there is nearly perfect registry of topographic and spectroscopic data. As a practical concern, the number of pixels in the scan or the scan area may be reduced to prevent piezo creep or thermal drift from moving the feature of study or the scan area during the duration of the scan. While most CITS data obtained on the times scale of several minutes, some experiments may require stability over longer periods of time. One approach to improving the experimental design is by applying feature-oriented scanning (FOS) methodology. ==Data interpretation==

From the obtained I-V curves, the band gap of the sample at the location of the I-V measurement can be determined. By plotting the magnitude of I on a log scale versus the tip-sample bias, the band gap can clearly be determined. Although determination of the band gap is possible from a linear plot of the I-V curve, the log scale increases the sensitivity. {{NumBlk||\frac{dI/dV}{I/V} = \frac{ \displaystyle \rho_s\left(r,eV\right)\rho_t\left(r,0\right) + \int_0^{eV} \frac{\rho_s\left(r, E\right)\rho_t\left(r,E-eV\right)}{T\left(eV,eV,r\right)}\frac{dT\left(E,eV,r\right)}{dV}\,dE }{ \displaystyle \frac{1}{eV} \int_0^{eV}\rho_s\left(r,E\right)\rho_t\left(r,E-eV\right)\frac{T\left(E,eV,r\right)}{T\left(eV, eV, r\right)}\,dE }\ .|}} Feenstra et al. argued that the dependencies of T\left(E, eV, r\right) and T\left(eV, eV, r\right) on tip-sample spacing and tip-sample bias tend to cancel, since they appear as ratios. This cancellation reduces the normalized conductance to the following form: {{NumBlk||\frac{dI/dV}{I/V} = \frac{d\left(\ln I\right)}{d\left(\ln V\right)} = \frac{\rho_s\left(r,eV\right)\rho_t\left(r,0\right)+A\left(V\right)}{B\left(V\right)}\ ,|}} where B\left(V\right) normalizes T to the DOS and A\left(V\right) describes the influence of the electric field in the tunneling gap on the decay length. Under the assumption that A\left(V\right) and B\left(V\right) vary slowly with tip-sample bias, the features in \left(dI/dV\right)/\left(I/V\right) reflect the sample DOS, ==Limitations==

While STS can provide spectroscopic information with amazing spatial resolution, there are some limitations. The STM and STS lack chemical sensitivity. Since the tip-sample bias range in tunneling experiments is limited to \pm\phi/e, where \phi is the apparent barrier height, STM and STS only sample valence electron states. Element-specific information is generally impossible to extract from STM and STS experiments, since the chemical bond formation greatly perturbs the valence states. At finite temperatures, the thermal broadening of the electron energy distribution due to the Fermi-distribution limits spectroscopic resolution. At T = 300\,\mathrm{K}, k_\text{B}T\approx0.026\,\mathrm{eV}, and the sample and tip energy distribution spread are both 2k_\text{B}T\approx0.052\,\mathrm{eV}. Hence, the total energy deviation is \Delta E\approx0.1\,\mathrm{eV}. Assuming the dispersion relation for simple metals, it follows from the uncertainty relation \Delta x\Delta k\ge 1/2 that {{NumBlk||\Delta E\ge \frac{\hbar^2k_F}{2M^*\Delta x} = 0.47\frac{E_F-E_0}{rk_F}\ ,|}} where E_F is the Fermi energy, E_0 is the bottom of the valence band, k_F is the Fermi wave vector, and r is the lateral resolution. Since spatial resolution depends on the tip-sample spacing, smaller tip-sample spacings and higher topographic resolution blur the features in tunneling spectra. Despite these limitations, STS and STM provide 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. ==References==