The story so far: Isolated defects

Transcription

1 The story so far: Infinite, periodic structures have Bloch wave single-particle states, labeled by a wavenumber k. Translational symmetry of the lattice + periodic boundary conditions give discrete allowed values of k, and a modified free Fermi gas description is not unreasonable. Crystal structure dictates band structure + electronic occupation determines Fermi level. Doping can alter Fermi level for a given material. What happens in an imperfect material? Isolated defects dynamic defects Boundaries Isolated defects Strictly speaking, once symmetry of lattice potential is broken, Bloch waves are no longer exact single-particle eigenstates. Do we care? Not if defect density is low - don t expect entire electronic structure of a atom crystal to be changed very much by having a few defects. Crude definition of low: Define a typical distance between defects, l. If k F l >> 1, then Bloch waves are a reasonable description. If k F l << 1, then defects are spaced more often than effective wavelength of Bloch waves - poor description. 1

2 Types of defects - point defects Point defects: Substitution Interstitial Vacancy Schottky / Frenkel Point defects Some energetic cost associated with formation of defect, related to binding energy of solid. Different energy cost for defect motion. Defect motion usually thermally activated - some diffusive process, with D ~ exp(- /k B T). Usually lower costs to move defects along free surfaces: fewer atoms to reshuffle / satisfy chemically. ALWAYS some density of point defects, for free energy reasons. May act as scatterers for charge carriers. May act as traps for charge carriers. 2

3 Line defects - dislocations edge dislocation An extra row of atoms Propagation of dislocation allows easier deformation of crystal than bulk slippage - lump under rug analogy. Dislocation line moves in same direction as shear. Dislocations Screw dislocation Has chirality. Propagation of dislocation allows easier deformation of crystal than bulk slippage - tearing phonebook analogy. Dislocation line moves perpendicular to shear. 3

4 Dislocations May also act as electron scatterers like point defects Largely responsible for plastic mechanical properties of (poly)crystalline solids. Surfaces + elastic fields play a major role in dislocation motion - work hardening, etc. Weirdness at nanoscale: superplasticity - grain boundary migration enhanced Lu et al., Science 287, 1463 (2000). Interfaces What happens at actual material interfaces? Two big cases: Boundary between material and vacuum Boundary between one type of material and another In first case, because of break in lattice symmetry, expect Bloch wave picture to break down somehow, though still be good far from interface. Second case is much more complicated. 4

5 Work function Definition: the amount of energy required to take an electron from the Fermi level within a material and remove it to infinity. Usually assume a particle at infinity has an energy of zero. Then we find that E F, the energy of the highest occupied single particle state in the material, must be negative. Charge density at surface of crystal is not identical to that within the crystal. Work required to remove an electron through the surface layer is W s. Result: Φ, the work function, = difference between energy at infinity and energy of bound electron = -E F +W s. Work function Why is there a surface layer effect? Charge spills out into empty space, resulting in a dipole layer pointed toward the metal. Surface layer effect can depend on particular crystal face and polarity of bonding. Single atomic layers of junk can strongly affect Φ by altering the surface charge layer. In practice, work function is measured empirically - photoemission, thermionic emission. Li Au 2.4 ev ~ 5 ev 5

6 Surface states Spillage of electrons across interface implies bound states tied to the interface. General idea: interruption of background charge from ion cores leads can lead to bound states at the interface. Tamm states - general consequence of breaking periodic potential. Label surface normal z. Surface states are bound in z, but may be free in x and y. Disorder (impurity, unsatisfied chemical bond, vacancy) can lead to surface states that are localized in all 3 directions. Surface states may be empty or full, depending on Fermi level. Surface states Image from IBM-Almaden Occupied free surface states of Cu (111), confined by a ring of iron atoms. 6

7 Tamm states Remember the Kronig-Penney model? V(x) b x Tamm model: V 1 V(x) -V 0 a b x -V 0 a Tamm states Solving the K-P model in a crystal with periodic boundary conditions led to bands of allowed states, each with real values of k. Those allowed single-particle states are Bloch waves, and are free or delocalized in the sense that the wavefunctions extend throughout the sample. Tamm model: to meet b.c., k has to become complex, allowing wavefunction to exponentially decay away from surface. Results: For large crystal, complex part of k is small for states in the middle of a band; decay length is long compared to crystal size. These states are relatively unaffected. A new state appears, one for each band, in the gap. State has large complex component of k, and is spatially localized at sample edge. 7

8 Surface states summary Surface states exist when symmetry of infinite periodic lattice is broken. Surface states can be localized or delocalized in the x-y plane, and may be empty or full depending on position of E F. Density of surface states may be quite significant - in 1d Tamm model, one (z-confined) per longitudinal band. Result for higher dimensional case: number of surface states can be comparable to number of atoms on surface! Nanoscale surface state issues: number of surface states can be comparable to number of bulk states in small particles! Joining materials with different Fermi levels Can get surface states. Can get appreciable charge transfer, causing local changes in band structure! Many energy scales to contend with: E F, E C, E V, Φ on both sides of junction, etc. Truly complete 1 st principles picture still doesn t exist. Certain common situations arise in technology. 8

9 Joining materials with different Fermi levels Start with two metals with different work functions: E F E F What happens when they re brought together? Electrons flow from system of higher chemical potential to that of lower chemical potential. Charge transfer: How much charge? Total electrochemical potential (including voltage!) must end up being uniform across junction. Bands effectively bend because of double charged layer at interface. Double charge layer because departing electrons leave behind ion cores. Thickness of charge layer called depletion width ; atomic scale in metals Conventional way of drawing: shift bands to allow E F to be uniform across sample. 9

10 Charge transfer: Why don t we see the effects of these charge layers all the time? Dipole layer in metals is so thin it doesn t affect transport. Going around a closed circuit voltages must sum to zero, so this is very challenging to measure directly. Remember, diffusive equilibrium really requires µ/t to be constant. Temperature gradients can lead to detectable voltage shifts: the thermoelectric effect. What about other material combinations? Metal-semiconductor junctions Before contact: vacuum E F E C E V After contact: Schottky barrier! Depletion width much larger than in metal case

11 Metal-semiconductor junctions Schottky barrier! Schottky barrier makes it difficult to inject electrons from metal into semiconductor. Result: nonlinear IV behavior diode. Very small barriers can result in (almost) Ohmic contact. Very much an art form. Schottky barriers used in a number of devices. Semiconductor-semiconductor junctions Many possibilities exist. One of the most commonly used: the p-n junction p-type n-type Depletion region 11

12 Semiconductor-semiconductor junctions Another commonly used structure: GaAs/AlGaAs interface Molecular electronics? Contacts in molecular electronic devices remain poorly understood. Charge transfer at the interface with metals undoubtedly plays an important role. Remember, while the charge layer length scales are often relatively small in metal-metal joints, for example, even a two-atom thick space charge layer can be considerable in a small molecule! 12

13 Summary Many types of structural defects are possible most small ones (point, line) don t have profound effects on electrons in large systems. Interfaces can have big effects. Surface states can exist and be comparable in number to bulk states in small particles. Charge transfer generally takes place at the interface between dissimilar materials, with big consequences for local band structure. Next time Chemical origins of band structure 13