Electron tunneling transmission probability

In contrast to the discussion above with amorphous barriers, it is possible to use first-principles electron-structure calculations to describe TMR with crystalline tunnel barriers. In the Julliere model the TMR is dependent only on the polarization of the electrodes, and not on the properties of the barrier. In contrast, theoretical work by Butler and coworkers showed that the transmission probability for the tunneling electrons depends on the symmetry of the barrier, which has a dramatic influence on the calculated TMR values [20]. In the case of Fe(100)/Mg0(100)/Fe (100) the majority of electrons in the Fe are spin-up. They are derived from a band of delta-symmetry. In 2004 these theoretical predictions were experimentally confirmed by Parkin et al. and Yusha et al. [21, 22]. Remarkably, by 2005 TMR read heads were introduced into commercial hard disk drives. 

Thus the transmission probability T decays exponentially with barrier thickness L. Indeed, it decreases by a factor of 10 for every 0.1 nm of barrier thickness, indicating that electron tunneling is very short range indeed. 

Quantum mechanical tunneling is a result of the finite potential barrier at the metal-vacuum interface. The electronic wave function ip and its first derivative dip/dz are continuous across this interface (or finite potential discontinuity), the electron wave function decaying exponentially, e in the forbidden region where the barrier exceeds the total electron energy. In this context, k is approximately related to the apparent work function or mean local tunneling barrier, = 2m((p — E)/ h/2nf-. Thus, the tunneling current. It, or transmission probability also decays with barrier width, z [6.10-6.13]... 

For a specific combination of electron donor and acceptor, the rate of electron tunneling is proportional to the transmission probability, with/c = 7 nm" (eqn 9.11). By what factor does the rate of electron tunneling between two co-factors increase as the distance between them changes firom 2.0 nm to 1.0 nm ... 

The tunneling current density is determined by the density of electrons per unit time incident on the junction and the transmission probability as a function of energy. Integrating over all electron energies one obtains " ... 

The probability matrix plays an important role in many processes in chemical physics. For chemical reactions, the probability of reaction is often limited by tunnelling tlnough a barrier, or by the fonnation of metastable states (resonances) in an intennediate well. Equivalently, the conductivity of a molecular wire is related to the probability of transmission of conduction electrons tlttough the junction region between the wire and the electrodes to which the wire is attached. 

In tunneling electron transfer, the transfer rate Vt at the energy level e can be given by the product of the state density occupied by electrons in the initial state, the state density vacant for electrons hi the final state of the step, the tunneling probability Wt, and the transmission coefficient K( as expressed in Eqn. 7-30 ... 

From this presentation it is seen that the overall probability P of electron transfer is approximated by P = Pn Pe, where Pn is the probability that the nuclei achieve a configuration so that the electron energy is identical for the initial and final systems (Fig. 5b). In usual kinetic terms, this means that the rate constant is given by kf = /r(kp )ad, where k is the transmission coefficient k = 1 for an adiabatic electron transfer k < 1 for nondiabatic transfers) and (k ) is the rate constant observed for an adiabatic electron transfer. The latter then depends only on nuclear motions that affect the potential energy of the electron. In usual chemical terms, (kf ) is directly related to the height of the activation barrier, that is, to the energetic separation between the state where the electron may tunnel and that corresponding to the initial system at equilibrium (compare Fig. 5b), denoted [A, D] in Scheme 3. 

Figure 2 Schematic diagram of a one-dimensional tip sample junction. A positive bias is applied to the sample. Thus, electrons can tunnel from occupied tip states into unoccupied sample states. The lengths of the horizontal arrows indicate the different transmission coefficients (tunneling probability) for electrons with different energies.

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