Scanning tunneling spectroscopy and local density of states

Choose positive sample bias V to access unoccupied sample states, and define positive tunneling current for that electron-transfer direction. Measure sample energy E from its chemical potential, with the tip chemical potential higher by eV. In weak elastic tunneling, a golden-rule expression combines sample and tip densities of states, the squared tunneling matrix element, and the difference of their occupation factors. Energy conservation selects pairs of states at the same absolute energy, not simply at the same energy measured from their separate chemical potentials.

With a featureless tip and nearly constant matrix element over the relevant energy window, the zero-temperature positive-bias current integrates the sample's local density of states from zero to eV. Differentiation then selects its upper boundary. This useful limit is often summarized as conductance measuring local density of states, but the local quantity is evaluated through the tip wave function and barrier. Tip orbital structure, a superconducting or magnetic tip, inelastic channels, and bias-dependent transmission change the relation.

In constant-current imaging a feedback loop changes tip height to hold an integrated tunneling current at its setpoint. More accessible sample spectral weight can therefore make the tip retract even when the atomic positions are unchanged. The approximate exponential dependence on tip distance amplifies small geometric changes, but it also makes the current sensitive to the barrier and tip condition. A bright topographic feature can combine true height, energy-dependent electronic structure, and local work-function changes.

For spectroscopy the feedback is commonly stabilized at a chosen bias and current, then disabled while a bias trace is recorded at approximately fixed height. Different stabilization conditions can give different multiplicative spectral weights. Comparing maps therefore requires tracking the setpoint and testing reproducibility with several tips and heights. Normalized conductance can reduce some matrix-element variations, but division by current or voltage can introduce its own singular or misleading behavior near zero current and does not universally yield an absolute density of states.

At finite temperature the derivative of the Fermi function broadens a sharp feature. Lock-in modulation, voltage noise, instrumental response, and intrinsic lifetime broadening add further kernels. Their widths cannot always be subtracted by a Gaussian rule. A suppressed zero-bias conductance can indicate a gap in accessible local spectral weight, but a tunneling matrix-element minimum, junction instability, or charging effect can also suppress current. Spatial reproducibility and temperature or field dependence help distinguish these mechanisms.

Fourier transforming conductance modulations can reveal quasiparticle interference: defects connect states by scattering wave vectors. The map depends on the impurity potential, orbital and coherence factors, and the energy-dependent electronic propagators. It is not generally a direct image of the Fermi surface or a joint density of states without weighting. A missing scattering vector need not mean the corresponding states are absent. Spatially resolved spectra should be compared with diffraction and bulk measurements before surface inhomogeneity is assigned to an entire sample.