The story so far: Isolated defects

Transcription

1 The story so far: Infinite, periodi strutures have Bloh wave single-partile states, labeled by a wavenumber k. Translational symmetry of the lattie periodi boundary onditions give disrete allowed values of k, and a modified free Fermi gas desription is not unreasonable. Crystal struture ditates band struture eletroni oupation determines Fermi level. Doping an alter Fermi level for a given material. What happens in an imperfet material? Isolated defets dynami defets Boundaries Isolated defets Stritly speaking, one symmetry of lattie potential is broken, Bloh waves are no longer exat single-partile eigenstates. Do we are? Not if defet density is low - don t expet entire eletroni struture of a atom rystal to be hanged very muh by having a few defets. Crude definition of low: Define a typial distane between defets, l. If k F l >>, then Bloh waves are a reasonable desription. If k F l <<, then defets are spaed more often than effetive wavelength of Bloh waves - poor desription.

2 Types of defets - point defets Point defets: Substitution Interstitial Vaany Shottky / Frenkel Point defets Some energeti ost assoiated with formation of defet, related to binding energy of solid. Different energy ost for defet motion. Defet motion usually thermally ativated - some diffusive proess, with D ~ exp(- /k B T). Usually lower osts to move defets along free surfaes: fewer atoms to reshuffle / satisfy hemially. ALWAYS some density of point defets, for free energy reasons. May at as satterers for harge arriers. May at as traps for harge arriers.

3 Line defets - disloations edge disloation An extra row of atoms Propagation of disloation allows easier deformation of rystal than bulk slippage - lump under rug analogy. Disloation line moves in same diretion as shear. Disloations Srew disloation as hirality. Propagation of disloation allows easier deformation of rystal than bulk slippage - tearing phonebook analogy. Disloation line moves perpendiular to shear. 3

4 Disloations May also at as eletron satterers like point defets Largely responsible for plasti mehanial properties of (poly)rystalline solids. Surfaes elasti fields play a major role in disloation motion - work hardening, et. Weirdness at nanosale: superplastiity - grain boundary migration enhaned Lu et al., Siene 87, 463 (). Interfaes What happens at atual material interfaes? Two big ases: Boundary between material and vauum Boundary between one type of material and another In first ase, beause of break in lattie symmetry, expet Bloh wave piture to break down somehow, though still be good far from interfae. Seond ase is muh more ompliated. 4

5 Work funtion Definition: the amount of energy required to take an eletron from the Fermi level within a material and remove it to infinity. Usually assume a partile at infinity has an energy of zero. Then we find that E F, the energy of the highest oupied single partile state in the material, must be negative. Charge density at surfae of rystal is not idential to that within the rystal. Work required to remove an eletron through the surfae layer is W s. Result: Φ, the work funtion, differene between energy at infinity and energy of bound eletron -E F W s. Work funtion Why is there a surfae layer effet? Charge spills out into empty spae, resulting in a dipole layer pointed toward the metal. Surfae layer effet an depend on partiular rystal fae and polarity of bonding. Single atomi layers of junk an strongly affet Φ by altering the surfae harge layer. In pratie, work funtion is measured empirially - photoemission, thermioni emission. Li Au.4 ev ~ 5 ev 5

6 Surfae states Spillage of eletrons aross interfae implies bound states tied to the interfae. General idea: interruption of bakground harge from ion ores leads an lead to bound states at the interfae. Tamm states - general onsequene of breaking periodi potential. Label surfae normal z. Surfae states are bound in z, but may be free in x and y. Disorder (impurity, unsatisfied hemial bond, vaany) an lead to surfae states that are loalized in all 3 diretions. Surfae states may be empty or full, depending on Fermi level. Surfae states Image from IBM-Almaden Oupied free surfae states of Cu (), onfined by a ring of iron atoms. 6

7 Tamm states Remember the Kronig-Penney model? V(x) b x Tamm model: V V(x) -V a b x -V a Tamm states Solving the K-P model in a rystal with periodi boundary onditions led to bands of allowed states, eah with real values of k. Those allowed single-partile states are Bloh waves, and are free or deloalized in the sense that the wavefuntions extend throughout the sample. Tamm model: to meet b.., k has to beome omplex, allowing wavefuntion to exponentially deay away from surfae. Results: For large rystal, omplex part of k is small for states in the middle of a band; deay length is long ompared to rystal size. These states are relatively unaffeted. A new state appears, one for eah band, in the gap. State has large omplex omponent of k, and is spatially loalized at sample edge. 7

8 Surfae states summary Surfae states exist when symmetry of infinite periodi lattie is broken. Surfae states an be loalized or deloalized in the x-y plane, and may be empty or full depending on position of E F. Density of surfae states may be quite signifiant - in d Tamm model, one (z-onfined) per longitudinal band. Result for higher dimensional ase: number of surfae states an be omparable to number of atoms on surfae! Nanosale surfae state issues: number of surfae states an be omparable to number of bulk states in small partiles! Joining materials with different Fermi levels Can get surfae states. Can get appreiable harge transfer, ausing loal hanges in band struture! Many energy sales to ontend with: E F, E C, E V, Φ on both sides of juntion, et. Truly omplete st priniples piture still doesn t exist. Certain ommon situations arise in tehnology. 8

9 Joining materials with different Fermi levels Start with two metals with different work funtions: E F E F What happens when they re brought together? Eletrons flow from system of higher hemial potential to that of lower hemial potential. Charge transfer: ow muh harge? Total eletrohemial potential (inluding voltage!) must end up being uniform aross juntion. Bands effetively bend beause of double harged layer at interfae. Double harge layer beause departing eletrons leave behind ion ores. Thikness of harge layer alled depletion width ; atomi sale in metals. Conventional way of drawing: shift bands to allow E F to be uniform aross sample. 9

10 Charge transfer: Why don t we see the effets of these harge layers all the time? Dipole layer in metals is so thin it doesn t affet transport. Going around a losed iruit voltages must sum to zero, so this is very hallenging to measure diretly. Remember, diffusive equilibrium really requires µ/t to be onstant. Temperature gradients an lead to detetable voltage shifts: the thermoeletri effet. What about other material ombinations? Metal-semiondutor juntions Before ontat: vauum E F E C E V After ontat: Shottky barrier! Depletion width muh larger than in metal ase.

11 Metal-semiondutor juntions Shottky barrier! Shottky barrier makes it diffiult to injet eletrons from metal into semiondutor. Result: nonlinear IV behavior diode. Very small barriers an result in (almost) Ohmi ontat. Very muh an art form. Shottky barriers used in a number of devies. Semiondutor-semiondutor juntions Many possibilities exist. One of the most ommonly used: the p-n juntion p-type n-type Depletion region

12 Semiondutor-semiondutor juntions Another ommonly used struture: GaAs/AlGaAs interfae Moleular eletronis? Contats in moleular eletroni devies remain poorly understood. Charge transfer at the interfae with metals undoubtedly plays an important role. Remember, while the harge layer length sales are often relatively small in metal-metal joints, for example, even a two-atom thik spae harge layer an be onsiderable in a small moleule!

13 Summary Many types of strutural defets are possible most small ones (point, line) don t have profound effets on eletrons in large systems. Interfaes an have big effets. Surfae states an exist and be omparable in number to bulk states in small partiles. Charge transfer generally takes plae at the interfae between dissimilar materials, with big onsequenes for loal band struture. Next time Chemial origins of band struture 3

14 Chemial approahes to eletroni struture So far, we ve approahed eletroni struture by starting from a free eletron gas and tweaking it: Free eletron gas Find single-partile states ompatible with boundary onditions. States labeled by k, related to spatial periodiity of planewave. Certain disrete k values allowed. Energy paraboli in k. Nearly free eletron gas Single-partile states are Bloh waves. States labeled by k, but k is multivalued in E - bands! Certain k values allowed; speial things happen when k related to spatial harmoni of lattie. Energy paraboli in k near extrema - effetive mass. Before we go further, let s look at this from the hemial perspetive. Eletroni struture problems Small moleules: Maromoleules/ Nanorystals: Bulk solid: Si Si - atoms May be solved exatly (within various approximation shemes) s of atoms Bulk struture ~-5% surfae 3 atoms ioni/ovalent/metalli bonds ~% surfae ow do we approah this ase? Will show one simple method that gives all the right answers qualitatively (though it s lousy for serious quantitative alulations).

15 ukel theory: Linear ombinations of atomi orbitals Unsurprisingly, the key is to pik an approximation method that makes the problem tratable. ukel, 93s, ame up with LCAO, a variational tehnique. Consider ethylene as a speifi example: Available atomi orbitals for arbons: s s p We have 4 valene atomi orbitals to play with: s, p x, p y, and p z. First, hoose linear ombination of these instead: C C Three sp orbitals One π orbital ukel theory: Linear ombinations of atomi orbitals Let s assume that the ovalent bonds to the hydrogens are taken are of by the sp orbitals, and only worry about the C-C bond. C C Trial wavefuntion: Assume, are real onstants. The true (single-partile) ground state minimizes under variations of the s. E The amiltonian: V e Ve inter

16 3 Write the variational funtion: ( ) ( ) ( )( ) C C Assume that and are orthogonal (not great if atoms very lose). Define: Generally assume nonzero only for nearest neighbors. Then: Now onsider varying eah independently and saying that the variation in <E> must. E ( ) E δ δ ( ) ( ) ( ) E Doing the variation:

17 4 Doing the variation: C C E X X X, This is the ukel equation. The idea: find values of <E> that satisfy this, eah with its orresponding, pair. (Bonding and antibonding orbitals) ukel figured one ould alulate and for different pairs of atoms. For there to be nontrivial solutions, determinant of matrix must. on-site energy ovalent energy Solution for this ase: ±. E C C More general ase: Only nearest neighbors. Zero spatial overlap. Neglet sp framework. By onvention, set zero of energy so that. For typial C-C π bond, ~ -.75 ev. X X X X X L L L L Tridiagonal matrix Solution: os N j E j π

18 Density of states E j jπ os N Turns out if you atually solve this all the way to looking at the wavefuntions, the wavelength of a partiular j solution looks, in the limit of large j, like /atomi spaing (a). Rewriting in terms of k πj /(N )a, E k Density of states: ν os ( ka) / k ~ E a ( E / ) States lump here: E k os for. a π/a k Bands opefully now you an see where bands ome from in this piture. Eah band derives from a bonding orbital. Larger overlap integral larger bandwidth. E arrison 5

19 From bonds to bands What s the best language to use when desribing a many-atom problem? Answer evolves under irumstanes: arrison Big aveats: ukel theory essentially ignores symmetry requirements on wavefuntion - added in by hand at the end. Also ignores eletron-eletron interations. Stritly speaking, variational methods are only onstrained to give the best approximation to the lowest lying state with the symmetry of the orbitals used (generally only the ground state!). 6

20 Better eletroni struture methods: Tight binding (see notes from week ). artree-fok - trying to self-onsistently inlude eletroneletron interation effets. Loal density approximation Density funtional theory Time-dependent density funtional theory 7