SolidStatePhysics(PHY210)Vorlesung /Lectures:Wednesday13h00– 15h45Raum /Room:Y36-K-08http://www.physik.uzh.ch/lectures/fkp/Exam:Oral(mostlikely9-10th ofJune– detailstobeannounced)

JohanChangjohan.chang@physik.uzh.ch

Übungen /Exerciseclass:Fridays15h00– 15h45Raum /Room:Y36-K-08

DanielDestrazdestraz@physik.uzh.ch

StefanHolensteinstefan.holenstein@psi.ch

Literature

Google- power

Lastweeksexercise2solvedinchapter3(page71inmyversion)Lastweeksexercise3isexercise5ofchapter3inKittel.Solutioncanactuallybegoogled.(Lastweeksexercise1a,bwasbasicallysolvedduringthelecture.)

Today’slecture

Crystalbindingmechanisms Latticevibrations

CovalentCrystals

Example1

Example2

Example3

http://sciencenotes.org/periodic-table-showing-shells/

Periodictable

CovalentCrystals

C

1s22s22p2

C

1s22s2p3

+4eV

Diamondstructure

4covalentbonds

Electronicorbitals

1/2(ψs +ψpx +ψpy +ψpz)"

1/2(ψs +ψpx �ψpy �ψpz)#

1/2(ψs �ψpx +ψpy �ψpz)#

1/2(ψs �ψpx �ψpy +ψpz)#

1 ψ"

2 ψ"

3 ψ"

4 ψ"

ψpx ψpy ψpz ψs

Orbitalhybridization

Tetraeder

1 ψ"

3 ψ"

2 ψ"

4 ψ"

Orbitalhybridization

Tetraeder

1 ψ"

3 ψ"

2 ψ"

4 ψ"

Orbitalmoleculartheory:ExampleCH4(Methane)

Orbitalmoleculartheory:𝜎 and𝜋bonding

Graphene:𝜎 and𝜋bonding

Graphene

Today’slecture

Bindungstyp Beispiel Bindungsenergie (eV)

Ionisch NaCl

LiF

8.23

10.92

Van-der-Waals Ar

Kr

0.080

0.116

Kovalent Diamant

Si

7.36

4.64

Metallisch Na

Fe

W

1.13

4.29

8.66

Wasserstoff-Brücken

H2O

HF

0.52

0.30

Summary

Phonon– LatticeVibrations

http://www.chm.bris.ac.uk/webprojects2000/igrant/theory.html

E. Maxwell, Phys. Rev. 86, 235 (1952) and B. Serin et al., Phys. Rev. B 86 162 (1952))

Phononscanmakesuperconductivity

Phononscanconductheat

Linearchain-Models

Structurefactor:S=∑ 𝒆&𝒊𝒒𝒓𝒊𝒊

Madelungsconstant:𝜶 = 𝟐𝒍𝒏(𝟐)DistortionEnergy :𝑬 = 𝟎. 𝟓 ∗ 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 ∗ 𝜹𝟐

Phonondispersion:𝛚 =

LA=LongitudinalAcousticLO=LongitudinalOpticalTA=TransversalAcousticTO=TransversalOptical

LongitudinalandTransversePhonons

LA=LongitudinalAcousticLO=LongitudinalOpticalTA=TransversalAcousticTO=TransversalOptical

Acousticandopticalmodes

https://www2.warwick.ac.uk/fac/sci/physics/current/postgraduate/regs/mpags/ex5/phonons/

LA=LongitudinalAcousticLO=LongitudinalOpticalTA=TransversalAcousticTO=TransversalOptical

Numberofphononbranches

p=numberofatomsintheprimitivecell

3acousticbranches3p-3opticalbranchesTotal3pphononbranches

J. Phys.: Condens. Matter 24 (2012) 053202 Topical Review

Figure 3. Influence of temperature on phonons. (a) Phonon dispersion of aluminium at 80 K as in figure 2. (b) Comparison between thevarious temperature dependences of the phonon frequency at the LL point (indicated by a red arrow in (a)): the explicitly anharmonic (ah)shift (see text), the shift due to electronic (el) excitations and the shift due to quasiharmonicity (qh), i.e. influence of thermal expansion [24].

Figure 4. Correlation between the deviation from experiment forthe lattice constants and the bulk moduli. The results for the threedifferent exchange–correlation functionals LDA, GGA–PBE andGGA–PBEsol are shown in blue, orange and green, respectively.Reproduced with permission from [25]. Copyright 2011 SpringerScience + Business Media.

explicit anharmonicities usually impact thermodynamic dataparticularly at high temperatures, the influence of non-adiabatic interactions is very much system-dependent. A fewexamples of this currently intensively investigated topic [26]will be mentioned below; several others can be found in theliterature, e.g. [27].

In order to ensure a high numerical precision whencomputing the various free energy contributions, great careneeds to be taken to sufficiently converge the results. Sincea large number of parameters need to be optimized, efficientscaling procedures can be applied for this purpose [21]. Someof the most important aspects for phonon calculations are:

• For some elements (e.g. Cu) the grid size of theaugmentation charges needs to be increased well beyondstandard values in order to obtain a convergence ofthe Gruneisen parameter (volume dependence of phononenergies) to less than 1%.

• For some elements (e.g. Al) extraordinary high k-pointmeshes for the electronic integration are necessary.

Inappropriate k-point meshes can even yield unphysicalimaginary phonons in the vicinity of the 0-point.

• In the direct force constant method the supercell sizeis a critical parameter. In order to resolve the phonondispersion with sufficiently high precision (e.g. Pb) or toidentify small (Kohn) anomalies in the phonon spectra (e.g.Pt), the supercell size needs to be sufficiently large.

A high precision enforced in the phonon calculationsallows us to unambiguously assign the remaining errorsto (i) missing free energy contributions such as non-adiabatic contributions mentioned before and (ii) the xcfunctionals providing unique information regarding sourcesof their failing. Figure 2 shows that LDA overestimates theexperimental data in most cases, while GGA underestimatesit. This behaviour is surprisingly systematic [21] andconsistent with the performance of these functionals alreadyat T = 0 K (see figure 4): The overbinding of LDA and thecorresponding too-small lattice constant leads to a predictionof a stiffer material with a bulk modulus that is too largeas compared to experiments. The opposite correlation isobserved for GGA. The situation cannot simply be resolvedby using the experimental value for the lattice constant, sincethis results in an artificial inner pressure of the system. Evenif the same (experimental) lattice constant is used for both xcfunctionals, the corresponding difference in phonon energiesremains almost the same and only their order is reversed, i.e.LDA/GGA under/overestimates the experimental phonons,respectively. The only way out of this dilemma is thedevelopment of improved xc functionals. As can be seenin figure 4, PBEsol [28] is significantly reducing theover-/underbinding of LDA/GGA for non-magnetic metals.Since PBEsol, however, does not improve the descriptionof magnetic materials, which are the main objective of thispaper, we will not consider this xc functional in upcomingdiscussions.

The systematic behaviour of the xc functionals becomeseven more apparent in the heat capacities. They are obtainedfrom a second derivative of the free energy (calculated withequation (1)), which is most often the target quantity formaterials research. The heat capacity, however, provides amore sensitive response to even tiny errors in the free energy.

5

http://iopscience.iop.org/article/10.1088/0953-8984/24/5/053202

Phononsinaluminium

1THz=4.14…meV

Phononsindiamond