is increased by 1 Å (0.1 nm). Because of this, even when tunneling occurs from a non-ideally sharp tip, the dominant contribution to the current is from its most protruding atom or orbital. Tunneling between two conductors As a result of the restriction that the tunneling from an occupied energy level on one side of the barrier requires an empty level of the same energy on the other side of the barrier, tunneling occurs mainly with electrons near the Fermi level. The tunneling current can be related to the density of available or filled states in the sample. The current due to an applied voltage V (assume tunneling occurs from the sample to the tip) depends on two factors: 1) the number of electrons between the Fermi level EF and EF−eV in the sample, and 2) the number among them which have corresponding free states to tunnel into on the other side of the barrier at the tip. The higher the density of available states in the tunneling region the greater the tunneling current. By convention, a positive V means that electrons in the tip tunnel into empty states in the sample; for a negative bias, electrons tunnel out of occupied states in the sample into the tip. For small biases and temperatures near absolute zero, the number of electrons in a given volume (the electron concentration) that are available for tunneling is the product of the density of the electronic states ρ(EF) and the energy interval between the two Fermi levels, eV. Half of these electrons will be travelling away from the barrier. The other half will represent the electric current impinging on the barrier, which is given by the product of the electron concentration, charge, and velocity v (Ii=nev), . The tunneling electric current will be a small fraction of the impinging current. The proportion is determined by the transmission probability T, so . In the simplest model of a rectangular potential barrier the transmission probability coefficient T equals |t|2. Bardeen's formalism A model that is based on more realistic wave functions for the two electrodes was devised by John Bardeen in a study of the metal-insulator-metal junction. His model takes two separate orthonormal sets of wave functions for the two electrodes and examines their time evolution as the systems are put close together. Bardeen's novel method, ingenious in itself, solves a time-dependent perturbative problem in which the