Electronic structure of solids

Prof. Stanislav S. Fedotov, Prof. Dmitry Aksyonov

​

Materials Chemistry

Center for Energy Science and Technology

October 6^th, 2025

Why do we need electronic structure?

• Deeply understand bonding between atoms
• Electronic conduction and charge transport
• Optical, magnetic, mechanical, etc. properties
• Design of new materials?

Description of interatomic interaction in solids

2

Sir John Edward Lennard-Jones

1. Jones J. E. Proc. R. Soc. Lond. A106 (1924) 441–462; 463-477 (Articles by W.Bragg in the same vol)

2. Lennard-Jones J. E. Proc. R. Soc. Lond. A109 (1925) 584–597 (Articles by P.Dirac in the same vol)

1924: Ne and Ar

1925: Kr and Xe

Using RbBr interatomic distance and compressibility as well as Br⁻, Kr, and Rb⁺ refractivity, he derived for 32 ionic crystals:

1) Interatomic distance (2% accuracy for monovalent)

predicted for MgSe (5%)

2) Compressibilities

3) Crystal energy

Essential role of electrons in interatomic interaction

3

• 1897 discovery of electron by E.Wiechert and J. Thomson
• 1913 planetary model of atom by N. Bohr
• 1924 wave properties of electron suggested by Louis de Broglie and proved in 1927 in Davisson–Germer experiment
• 1926 Schrödinger equation - for wave functions and can describe electrons correctly

E=-13.6 eV

Time dependent SE (ψ - wave function, H - operator of total energy):

kinetic energy + potential energy

Why?

4

Quantum chemistry allowed description of matter from first principles without any information from experiment, except for fundamental constants, such as electron mass, charge and planck constant

INPUT:

​

• Atomic number: 3
• N atoms in unit cell: 1

​

OUTPUT:

​

• Metall, E_cohesive = 1.63 eV/atom
• bcc lattice a = 3.51 Å at room T
• Electronic structure and all properties corresponding to ideal crystal

The calculated properties can match experiment exactly! but … slight approximations are still required giving 1%-10% typical error.

What is electronic structure?

5

Wave function Ψ, also called orbitals, one orbital per each electron

How to deal with solid with infinite number of electrons?

What are the examples of solids (crystals)?

6

What is the main difference of crystals from molecules?

• Translational symmetry - the atomic structure is repeated infinitely in three dimensions
• The number of symmetries is limited

7

Problem: How to represent the infinite system?

Evgraf Fedorov, mathematician, crystallographer and mineralogist

derived 230 symmetry space groups The Symmetry of Regular Systems of Figures,1891

Fourier transform: functions of time

8

The Fourier transform F of a function f(t) is a function F(ω) in frequency domain,

and is defined as:

Fourier transform of cos(ω₀t):

Fourier transform: functions of space

9

The Fourier transform F of a function f (r) is a function F (g) in g space domain:

Brillouin zone and Wigner seitz cell

10

• Primitive cell - cell with minimal volume, infinite number of possibilities
• First Brillouin zone (BZ) is a primitive cell in reciprocal space with the symmetry of the reciprocal lattice
• Wigner-seitz cell - the primitive cell with the symmetry of the Bravais lattice in direct lattice

First BZ for FCC lattice

11

• The reciprocal lattice for FCC lattice is BCC
• Г is the center of Brillouin zone
• High-symmetry directions are called with Greek letters (see here), high symmetry points with Latin
• Fundamental domain of BZ is often called irreducible Brillouin zone (IBZ)

​

More Brillouin zones

12

First 27 BZ for 2D square lattice

First 4 BZ for 3D lattices:

All BZ are of equal volume

Electrons feel periodic potential of ions

​

​

13

where G is a set of vectors and the V_G are Fourier coefficients

Bloch theorem for periodic systems

14

• k - new quantum number, vector in reciprocal space!
• n is band number from the solution of reduced spectral problem with PBC
• only one reciprocal cell -> finite volume problem
• e^ikr - invariant with respect k = k+G, where G is translation vector

Theorem: In periodic system, one-electron wavefunction

can be chosen to be a plane wave times the periodicity of the Bravais lattice:

Example of band structure for graphene

15

Glustion Fig D2

BZ and IBZ

one-dimensional cross-sections is the most common way to visualize bands

Change of energy along high-symmetry direction

Example of band structure for silicon

16

Ground state and Fermi energy (surface)

17

The ground state of N electrons is obtained by filling one-electron bands with energies ε_n(k) up to the Fermi energy. Some bands are fully filled, the others are empty

No Fermi surface for band gap materials! -> definition for metals

Case 1: The band is partially filled or overlapped

Case 2: The band is either completely filled or empty

Band formation

18

Copper band structure

Simple picture for band formation in quantum wells (Kronig-Penney model)

19

Finite barrier:

small splitting in energy between states due to coupling

​

Infinite barrier between two wells:

degenerate states

​

D. Snoke /Solid state physics

States with more nodes (shorter wavelength) will have higher

energy, while states with fewer nodes will have lower energy.

1D case, infinite chain of H atoms with non-interacting electrons

20

a

At k = 0, also known as Г

At k = π/a, also known as X

a

a - interatomic distance,

𝜙 - atomic orbital

The same procedure for intermediate k values

Band structure for hydrogen chain

21

Dronskowski

in-phase, bonding combination of all

atomic orbitals

out-of-phase, and the interaction is ultimately antibonding

nonbonding

First Brillouin zone in 1D case

= pi/a

pi/2a

Density of states

22

Cu [Ar] 4s1 3d10 FCC lattice with 1 atom 11 Kohn-Sham electrons

The parabolic behaviour in L-Г-K region resembles free electron gas, however it is interrupted by spaghetti-like d states

The red discs are from the experimental angle-resolved photoemission data

​

from Giustino

Band structure for free electron gas

Charge density

23

Isousrface of charge density for Silicon

2D section Charge density for Silicon

Discussion

24

• What is the difference of Brillouin zone from other primitive cells in the reciprocal space?

• Why antisymmetric solution has higher energy compared to the symmetric one?

• What is the difference of real crystal from ideal periodic crystal?

Individual studies:

• Reading
• R. Hummel, Electronic Properties of Materials
• A. Sutton, Electronic Structure of Materials
• C. Kittel, Introduction to solid state physics

​

25

Thank you for your attention!

26