Effective masses in semiconductors

Transcription

1 Effective masses in semiconductors The effective mass is defined as: In a solid, the electron (hole) effective mass represents how electrons move in an applied field. The effective mass reflects the inverse of the band curvature: the lower m*, the larger is the curvature. Flatter bands have larger effective mass; electrons are more accelerated. In a three dimensional band structure, the effective mass is a tensor.

2 Metal-semiconductor interfaces When a semiconductor and a metal are brought into a junction, the unbalanced Fermi level tend to align by transferring charge depleting the charge carriers. Equilibrium: A barrier is formed: The Schottky Barrier. In n junction, the Schottky barrier amounts: In p junction, the Schottky barrier amounts: Shottky diode: M-S junctions exhibit a rectification behavior: large current exists under forward bias, while almost no current exists under reverse bias. Forward: Reverse: M-S contacts are needed in devices. It is necessary to reduce the effect of Schottky barriers, by matching work function to bands of the SC, or by reducing the depletion layer as much as possible by high doping

3 Metal-semiconductor interfaces: the charge depleted region FULL DEPLETION APPROXIMATION 1) Assuming that all the impurities are ionized at the depleted layer, the charge can be approximated by a constant profile along the whole region: 2) The electric field at the semiconductor is linear with the distance: 3) The potential increases with the squared distance: 4) The total potential change along the depleted region must equal the built-in potential minus the applied potential V a The depleted region amounts

4 p-n junctions A p-n junction consists of two semiconductor regions with opposite doping type Since thermal equilibrium implies that the Fermi energy is constant throughout the p-n diode, the built-in potential equals the difference between the Fermi energies, E Fn and E Fp, divided by the electronic charge. I Holes want to diffuse into n-part. Electrons want to diffuse into p part. A electric field is formed at the interface, responding to that inhomogeneity in the carriers charge. This creates a bump in the electrostatic potential in order to annihilate the net flux of particles across the interface in thermal equilibrium. The result is that the electrochemical potential, however, is constant along the crystal.

5 p-n junctions Reverse Bias: Semiconductor p-n junctions show rectification behaviour. Forward Bias:

6 p-n junctions: The depleted region FULL DEPLETION APPROXIMATION 1) Assuming that all the impurities are ionized at the depleted layer, the charge can be approximated by a constant profile along both depleted regions. The larger the doping the shorter the length. 3) The linear electric field can be obtained from Gauss law 2) The charge at every side must be equal 4) The potential Now changes along the two depleted regions. The total potential must balance the built-in potential. So:

7 Heterogeneous p-n junctions Heterogeneous materials combine different impurity density with different band gap width. The built-in potential associated with the junction is still the difference between the Fermi levels. A quantum well can be formed by accumulating carriers at an intermediate layer with smaller band gap. Junction break-down The maximum reverse bias voltage that can be applied to a p-n diode is limited by breakdown. Breakdown is characterized by the rapid increase of the current under reverse bias. Quantum well Avalanche breakdown Tunnel breakdown: Quantum mechanical tunnelling of carriers through the band gap. The tunnel probability equals Where the electric field E= E g /(ql).