band theory of the electronic properties of solids 6

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4th COMPUTER CALCULATIONS ab initio band structure

Calculations of the electronic structure of solids are a spectacular example ofapplication of computers to the physical calculations. The term "the firstprinciples" (Latin ab initio ) indicates that an input to the calculation example forcopper is basically just the element atomic number Z = 29 , and the universalphysical constants. As a result of such calculations we get that copper is ametal cubic fcc structure, and not the other densities, elastic properties, andeven it can be deduced that copper is red. calculations using the generalformalism of band theory (presented in Chapters 2 and 3 ). In addition, requirethe introduction of new theoretical concepts. samouzgodnienia principle enablesiterative finding such solutions Schrodinger equation (energy values and wavefunctions) as well as periodic potential, which is needed as the size of input to

such calculations. Adding to the Hamiltonian exchange-correlation potentialcan,in part at least, include the effects of electron-electron interactions.Approximations are also needed an accounting nature, enabling the efficientexecution of computations. The whole calculation performs very complexcomputer program (in the order of several thousand lines of code), but feasibleon an ordinary personal computer. The Department of Physics and NuclearTechniques such calculations lead prof. And Dr. Stanislaw Kaprzyk. JanuszToboa.

4.1. PRINCIPLE ZAMOUZGODNIENIA (BASED ON A SINGLE ATOM).

For the hydrogen atom Schrdinger equation can be solved analytically. For theatoms of other elements to be taken into account the impact of the electronwith other electrons forming such a core and shell of the atom valenceelectrons. A rigorous solution of the Schrdinger equation taking into accountthe electron-electron interaction requires the use wieloelektronowej wavefunction ( r 1 , r 2 , r 3 , .., s 1 , s 2 , s 3 ..), dependent on the spatial

coordinates r and and spin s and all electrons. Extremely difficult numericalcalculations using this function has been performed only for a small number ofatoms, electrons, such as He, Li and Be. Already in 1928, Hartree proposedapproximate calculations of the electronic structure of atomswieloelektronowych. Hartree method relies on describing the motion of electronsusing jednoelektronowych wave functions. We assume that each of theelectrons of an atom moves in a complex accident potential, the potential of the

nucleus and the electron cloud potential Vel (r) . The potential Vel (r) can becalculated from electrostatics rights, if we know the wave functions of theother Z - 1 electrons. starting point for the calculation of wave functions arecalculated for the Coulomb potential of the nucleus of charge ZQ e . The sum ofthe probability density of finding the electron at a given point in space r ,

(4.1)

What is the theory of band?

Solution to the Schrdinger equationfor a p eriodic potential

Populating ba nds by electrons.metals and non-m etals

Computer calculationsab initio band structure

Principle samouzgodnienia

Exchange-correlation potential

Samouzgodnione calculateab initio structure oftheelectron crystal

The results of the example ofcopper

Extensions formalism:magnetism and the effects ofrelativistic

The evolution of the electronicstructure of the periodic table ofelementsin the crystalline state.Sum mary: po ssibilitiesand limits of band theory

Dictionary Pol-Eng-mat

Test questions

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.electron charge q e gives the electron charge density of the clouds.electrostatics With the right electrical potential can be calculated electroncloud. We assume the spherical symmetry of the electron density function, ien ( r ) = n (r). assumption of spherical symmetry n ( r ) is fully closed subshellstrue for the core and for orbital s . For not completely filled electron orbitalsp ,d, ... is a useful approximation to facilitate the implementation of electrostaticcalculations. This allows for efficient use of Gauss' law binding the electric fluxE (r) on the surface of a sphere of radius r (left-hand side of the equationbelow) with a cargo of electrons contained inside the sphere (right):

(4.2)

Electron electrostatic potential Vel ( r) is the integral of the field strength,

(4.3)

The resultant potential Schrdinger equation consists of a nucleus potential andthe potential of electron clouds,

(4.4)

For new potential need to calculate the new wave functions. Forkulistosymetrycznego potential wave functions in radial coordinates r , canbe written as the product of the feature functions

are eigenfunctions of the operator market. As such, only the angles are

functions , ie do not depend on the form of the potential and do notcontain any constants. These are known as the so-called eighteenth century.

spherical functions. To find R n, l ( r ) and energy E n, l numerically solve theSchrdinger equation efficiently .

(4.5)

with the potential U (r) calculated numerically from formulas (4.3) and (4.4).The total energy of an atom is equal to the sum of the energies of states filled.samouzgodnienia principle lies in the fact that the calculated wave functionscan be used to construct the new electron density. The sequence ofcalculations (equations 1.2 -> 1.3 -> 1.4 -> 1.5 -> 1.2 -> 1.3 ... etc.) are madeperiodically until the next iteration obtained wave functions and energyeigenvalues are not significantly different from the previous ones. Examplepioneering results of calculations for an atom wave functions are shown inFigure 4.1a sodium. The most important consequence of the inclusion qualitativepotential of the electron cloud is that the energies of states does not dependonly on the principal quantum number n (as in the Bohr model of the Schrdingerequation or Coulomb potential), but for the n increase with increasing quantumnumber l(Fig. 4.1b).
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Figure 4.1a. wave functions 1 s , 2 s , 2p and 3 s for the sodium atom. The graphshows the results of Hartree original? Ego. On the vertical axis postponed productrR ( r) where R ( r) is the radial part of the wave function. Fig 4.1b. Jednoczstkowe

energy levels for an atom of sodium and filling for 11 electrons. (Spin-orbit splitting3p level shows figure 4.10 ).
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4.2. POTENTIAL exchange-correlation

The original method is Hartree approximation, which overestimates the energyvalue jednoelektronowych states. This is because they do not take into accounttwo effects characteristic of quantum many-body system dynamics, known ascorrelation and exchange effects. A more detailed theory of the two effects ispresented in textbooks of quantum mechanics, here only a cursory attempt toexplain the nature of the two effects. CORRELATION EFFECT. For definitenesslet us examine how Hartree method considers two electrons move. Theassumption that one electron "feels" the presence of an electron by twospherical-symmetrical electron-electron density of 2 means that the probabilityof finding an electron at two points equidistant from the core is the same,regardless of the position of the electron 1 (Figure 4.2). The actual two-electron wave function is deformed in such a way that electrons remained as farfrom each other, as this decreases the energy of repulsion.

Figure 4.2. figure for secondary clarification and exchange correlation effects.

EFFECT OF EXCHANGE is an amazing phenomenon kwantowomechanicznym nothaving counterpart in classical physics. Holds for systems that do not changeunder the influence of operations in places Reset items identical particles.(Figure 4.2b shows the effect of the particles on the conversion operatorsystem shown in Figure 4.2a). Symmetry replacement surgery particles alsooccurs in classical physics, but that does not cause any effects. For example,the classic formulas describing the collision of balls of different masses are alsoautomatically correct for identical spheres. However, such a model describingthe Rutherford scattering of alpha particles (the nucleus 4 He) nucleus in anelectrostatic field is fair for all nuclei, from hydrogen to uranium, with the

exception of helium. The effect of the exchange causes the cross section forthe scattering of alpha particles on the 4 He is twice. effect the exchange of"not working" for electrons with different spins, such as electrons, are notidentical. This results in a further decrease while the energy of electrons withidentical spins. Slater in 1952 proposed a way to take into account theapproximate exchange and correlation effects in the Hartree method. Is to addan additional component to the Hamiltonian, which will be called exchange-correlation potential Vxc . A simple model
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(4.6)

was derived heuristically by Slater with a model of electrons interacting . Thevalue ofVxc at a point in space r depends only on the electron density n ( r )at that point what is called the local density approximation . (Electrostaticpotential of the electron cloud at the point r depends on the n ( r ), but in anon-local manner, since in order to calculate Velat a given point by the laws ofelectrostatics (4.2) and (4.3) need to know the distribution of the charge

density in the other sections) . overall potential, which we will use in thecalculation consists of electrostatic potential (nucleus and electron cloud) andthe exchange-correlation potential:

(4.7)

Inclusion of Vxc Schrdinger equation improves the compatibility experience

significantly compared to the original theory Hartree'go without increasing thecomputational difficulty. It can be used in the calculation of the electronicstructure of atoms, molecules and solids. since the theoretical concept ofexchange-correlation potential was established by the two theorems proved byHohenberg and Kohn in 1964. They can be formulated as follows:

(I) all the properties of the ground state of the electrons are interactingelectron density functionals,

(Ii) energy of the ground state which is in accordance with the previousstatement, the functional E { n (r )}, reaches a minimum value for the

true electron density.

Theorem (i), (ii) and a set of derived theoretical results is now called densityfunctional theory . Name highlights the fact that for the calculation of theproperties of interacting electrons do not need to know the wave function

wieloelektronowej! Just to know the electron density n ( r ) which is a functionof three spatial coordinates. The use of claim (ii) to solve the Schrdingerequation leads to the conclusion that there is a POTENTIAL V xc non-locallydependent on n ( r ), which is used in the Schrdinger equation would give theexact value of the energy of the ground state - but the theory does not specifyhow to construct! At the moment, to construct the exchange-correlationpotential is not used anymore Slater formula, but formula arising from the moreadvanced solutions for the electron gas model oddziaywujch. densityfunctional theory is an example of a fundamental result of non-relativisticquantum mechanics that has not been discovered in the early days of 1925-1930 but almost forty years later. This begs the question: is the theory ofquantum mechanics are hiding some important facts to be discovered?

4.3. CALCULATIONS SAMOUZGODNIONE ab initio electronicstructure CRYSTAL

Samozgodne band structure calculation carried out similar steps as described inthe case of an atom from the wave function, the electron density and theresultant value of the potential energy and the new wave functions.calculations begin by the adoption of a specific type of structure and latticeconstant. The initial value of the electron density from the previously calculatedatomic orbitals samouzgodnienia method, with the same approximation forexchange-correlation potential, which will then be applied to the crystal.Commonly used approximation accident on the spatial shape of the potential U(
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. ,cell is filled with touching spheres specific to atoms. The potential is assumedspherically symmetric inside the spheres and stood in the area between thespheres (Figure 4.3).

Figure 4.3 The potential cup-shaped. The graph shows

the potential along the course of the segment AB.

Zoom potential miseczkowego greatly facilitates the calculation by the fact thatthe potential depends on one variable - the distance from the nucleus. It is theexact equivalent of kulistosymetrycznoci assumptions about the potential of asingle atom. However, the solution to the Schrdinger equation is the hardestand most time-consuming part of the calculation. Developed at least a dozendifferent methods. Most of them are based on developing solutions in a numberof these and other basic functions, which brings the solution of differentialalgebraic Schrdinger equation to find the eigenvalues of the problem (ie,diagonalization) array of size depending on the number of atoms in the unit cell.(For example, for one of the methods on a 9 9 for Cu to 117 117 for high-temperature superconductor YBA 2 Cu 3 O 7 .) run different calculations for the

core electrons and valence electrons. Electron energy of the core is calculatedonce, for k = 0 For example, for copper need to be performed for the orbitals of1 s , 2 s , 2p , 3 s and 3p 18 filled by the core electrons. valence electrons forthe energy eigenvalues calculated need for a large number of values of k ,located at the edge, the side walls and the interior portion of the first irreducibleBrillouin zone. Chart dispersion relations for Cu (Figure 4.5) represents only partof the point E( k ) along selected sections on the reverse. All values ofE( k )stored in the computer's memory (eg, for 1410 points in the case of ourcalculations for Cu) will be used to construct a curve then the density of states(Fig. 4.6) and find the Fermi energy. Calculation ofD ( E) and EF need to do ateach step samouzgodnienia, because it is necessary to distinguish between filledand empty states. Only populated states use to construct the nextapproximation for the electron density. The calculation is repeated until thesamouzgodnienia (Fig. 4.4).


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Figure 4.4. diagram samouzgodnionych calculations of the electronic structure of the

crystal.

4.4. RESULTS ON THE EXAMPLE OF COPPER

Band structure calculation capabilities "ab initio" will be presented on theexample of copper. Unless otherwise indicated, the data and the results

presented in Chapter 5 are from the monograph "Calculated electronic structureof metals," published in 1976 by IBM physicists working in Moruzzi'ego, Janakaand Williams. For the first time then calculated using the same software, acollection of 32 electronic structure of metallic elements from a hypotheticalmetallic hydrogen (Z = 1) to indium (Z = 49).

Figures 4.5 and 4.6 show the basic characteristics of the electronic structure ofcopper dispersion relations E( k ) and the state density function D ( E) . (Formthe Fermi surface is shown in Figure 3.2 ). The Fermi energy determines thepoint at which the integral of the density of states (dotted line in figure 4.6)becomes equal to 11, ie, the number of valence electrons of Cu. Density ofstates at the Fermi level D ( E F ) is critical (but not exclusively) affect thevalue of the electron specific heat and other properties of the metal.
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Figure 4.5. relation E (k) for copper along selected sections of the Brillouin zone.Color parabo la show s, for comparison, the E (k) for electrons in the empty space.

(Marking points of W, L , X, K explains Fig. 3.2 ).

Figure 4.6. feature density of states. Dashed line represents the integral of the

D ( E).

Most of these values, we can now explore experimentally. In particular,dispersion relations and density of states function can be determined by meansof photoelectron spectroscopy, there are a number of methods for testing theFermi surface. Convincing demonstration of possible method is to set a series ofmacroscopic physical properties of Cu, possible to calculate the completeknowledge of the electronic structure.

Table 4.1. Selected results of calculations ab initio for Cuand their comparison with experiment.

Physical Size Theory Experiment

Type of network fcc fcc

Lattice constant [] 3.58 3.61

Crystal total energy [eV / atom on]566,82

-44

Total energy of the atom [eV]562,69

-44

Zero vibration energy [eV / atom] 0.04

Cohesive energy [eV / atom] 4.10 3.49

Compressibility modulus [GPa] 155 142
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erm energy ca cua e rom e o om o eband, in eV] 9.41

D ( EF ) [states / (eV atom)] 0.29 0.30

TYPE OF STRUCTURE. The calculation is carried out for a particular type andsize of the unit cell. Type of structure can, however, be calculated from firstprinciples based searching minimum total energy of the crystal lattice constantfor different structures (Figure 4.8). We find that the nature implemented bynetwork type Cu - surface-centered cubic (fcc) - provides a lower total energythan, say, the structure of space-centered (bcc) and simple cubic (sc).

Figure 4.8. principle identify the type o f structure, the lattice constant and

compress ibility module.

Minimum dependence E ( a ) defines the equilibrium value of a 0 FIXEDNETWORK. For Cu, the calculated value is in conformity to better than 1% ofthe value measured at 0 K. Matching the energy dependence of the volumeV = a 3 near the minimum,

(4.8)

allows to determine the theoretical value of volume compressibility modulus B .For the calculation of ab initio other elastic constants Cu need to performadequately accounts for the deformed networks, such as to obtainthe Young's modulus. cohesion energy is defined as the difference between the

total energy of atoms isolated and combined into a solid. The value 4.10 eV /atom is the energy difference between an atom and a small solid patch zerovibration energy (calculated using the Debye approximation as 9/8 kB T D ).the energy value of the first two have a decisive influence strongly associatedcore electrons whose energy in atoms and solids is similar (but not identical).Correct calculation of the cohesive energy, which is a small difference of twolarge numbers, require high precision calculations. (Using FORTRAN have to usethe so-called. Double precision). cohesive energy can be determined

experimentally by determining the heat required to carry out the crystal fromthe temperature of the gas phase zero. The higher the cohesion energy of thehigher melting point and boiling point.

4.5. EXTENDED formalism: MAGNETISM AND relativistic effects

Formalism of band theory were developed in different directions, the two

resented here are articularl im ortant modifications.
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a) magnetism

Calculations for crystals with arrangement of magnetic and ferromagneticantyferromagnetykw (so far only kolinearnych), are an extension of densityfunctional method. Instead of a single electron density n ( r ) in the calculationsof the electronic structure must take into account two different functions

describing the electron density of the spins directed "up" and "down".

Electrostatic potential is a function of the total electron densityand prefers the non-magnetic state. However, exchange-

correlation potential is different for electrons with different spins, and because itis negative (model 4.6) - was he "generates" instability in certain magneticmaterials. Figure 4.9 shows the function of the density of states for Fe, dividedinto separate parts corresponding to elect rons , filled up to the Fermi

energy common. The difference multiplied by the value of the Bohrmagneton can spin magnetic moment.


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Figure 4.9. Functions density of states for spin "up" and "down" for ferromagnetic

bcc iron.

Formalism outlined above allows you to calculate "the first principles" and thecrystal exhibits magnetic ordering and what is the value of magnetization (Table4.2). The basic "conversion" of the elect ronic structure affects the properties ofall the iron, in particular the lattice constant is greater, and the compressibilitymodulus and cohesive energy are smaller as compared to a hypothetical non-magnetic Fe (Table 4.2).

Table 4.2. Comparison of the calculated values of magnetic bcc iron with theresults of the calculations "magnetic" and the experimental values.

Non-magnetic Fe Magnetic Fe Experiment

type of network bcc bcc bcc

lattice constant [] 2.72 2.79 2.86

cohesive energy [eV / atom] 5.97 5.52 4.28

compressibility modulus [GPa] 306 217 220

magnetic moment [ B ] - 2.15 2.22

b) relativistic effects

The Schrdinger equation is the non-relativistic equation. As can be calculatedfrom the Bohr model 1s electron in the hydrogen atom "orbits" at a speed of1/137 the speed of light, which is the speed of uranium (Z = 92) increases thevalue of (92/137) c Thus, relativistic effects increase with Z. increasingcollection of electronic structures calculated in the book " Calculated ... " endsat other times (Z = 49) , precisely because the authors usednierelatywistycznego algorithm, which fails for large Z. Currently relativisticcalculation methods are well controlled and electronic structure calculations canbe carried out for any heavy elements. Relativistic effects in band theory can bedivided into quantitative and qualitative. The first consists in changing theshape of the bands, which in the Schrdinger equation can be achieved byadopting a relativistic expression for the kinetic energy Hamiltonian. The resultis, among others. lower energy bands originating from the electron p (S-wavefunctions are closer to the nucleus (see Figure 4.1a ), which in an image Bohrmeans that achieve a great speed, which leads to an increase of the relativisticmass, which in turn to increase the binding energy). Color of gold (Z = 79) canbe considered as relativistic effect because Non-relativistic calculations predictsilvery color, while the relativistic calculation can explain the yellow color of themetal. 's qualitative relativistic effect is the splitting of the bands (those whichare non-relativistic theory of degenerate, see chap. 2.3e ) as a result of theso-called. spin-orbit interaction. Figure 4.10 resembles the effect of thisinteraction on the atomic level p in the non-relativistic theory of p contains 6

levels of electronic states with orbital quantum numbers l = 1 ( m l = -1, 0, 1)and spin s = 1/2 ( m s = 1/2). effect of spin-orbit interaction is the cleavageof the 4-fold degenerate levelj = 3/2 and the double-degenerate levelj = 1/2( j- the total angular momentum quantum number equal to l s ). Fission isobserved both in atomic spectra (eg sodium yellow line) and X-ray spectra(splitting the line into a doublet ). Much the same is at the top of the valenceband of silicon (Z = 14). Non-relativistic calculations ( Figure 2.7 ) provide thatcombine in a 3 subband, so the state k = 0 (point ) is 6-fold degenerate. Theeffect of spin-orbit interaction is shown in Figure fission 4.10b. Subbands,"heavy hole" and the "light hole" are combined at a point . (The names comefrom the subband effective mass value, equal to 0.52 and 0.16). However, thetip of subbands "split off" to a lower energy of = 44 meV. strong relativisticeffects increase with the increase of causes fission value for GaAs (Z = 31 and33) increases the value of = 340 meV.
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Figure 4.10. effects of spin-orbit interaction in (a) the atomic level 3 are sodium, (b)Si band structure near the top of the valence band.
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band theory of the electronic properties of solids 6

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